Catalog vs Lean — scientific opinion, divergence, human votes
Proof library
Dedicated home for Li's proof corpus — separate from benchmark wall-clock ratios. Compare scientific catalog opinion (TOML / proof-db) vs Lean scan, then vote what you believe.
Each classical-math row shows the LaTeX formal statement (from Lean), then source for Lean, Li, and TOML catalog. Use Export PNG on an expanded row to share on X.
Erdős #1 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #1 (partial): if A ⊆ {1..N} has n elements with…
proved (catalog)provedproved
Erdős register row — catalog marks proved; not claimed proved in Lean unless both votes agree.
Erdős #2 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full bounded minimal modulus of covering systems is literature (Hough/Balister); this pack closes…
proved (catalog)provedproved
Erdős register row — catalog marks proved; not claimed proved in Lean unless both votes agree.
Erdős #3 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — If A⊆ℕ has ∑_{n∈A} 1/n=∞, must A contain…
proved (catalog)provedproved
Erdős register row — catalog marks proved; not claimed proved in Lean unless both votes agree.
Erdős #4 (partial): prime witnesses + gap-shape scaffold. Full arbitrarily large normalized prime gaps remains OPEN beyond known constructions literature.
proved (catalog)provedproved
Erdős register row — catalog marks proved; not claimed proved in Lean unless both votes agree.
Erdős #1 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #1 (partial): if A ⊆ {1..N} has n elements with distinct subset sums then 2^n ≤ n·N+1 (hence N
Erdős #10 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist a set A of positive integers with ∑ 1/n = ∞ containing no three-term arithmetic pr
phase16 iter1323 shard1: witness→proved via Bloom–Sisask (`Li.ProofDb.ErdosMathlib.e_10_bloom_sisask_divergent_reciprocal_implies_3ap`); divergent reciprocal sum ⇒ 3-term AP; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_10_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #1000 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A=//{n_1<n_2<//cdots//}$ be an infinite sequence of integers, and let $//phi_A(k)$ count the
phase16 iter1242 shard1: witness→proved via Haight/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1000_phi_a_cesaro_mean_can_vanish`); exists strictly mono A with Cesàro mean of φ_A(k)/n_k → 0; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1000_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1001 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $S(N,A,c)$ be the measure of the set of those $//alpha//in (0,1)$ such that//[//left//lvert /
phase16 iter1284 shard2: witness→proved via EST/Kesten–Sós ax-wrap (`Li.ProofDb.ErdosMathlib.e_1001_erdos_szusz_turan_kesten_sos_diophantine_density`); Diophantine density limit (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1001_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #1002 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For 0<α<1, does f(α,n)=(1/log n)Σ_{k=1}^n(1/2-{αk}) have an asymptotic distribution? Proved parti
Erdős #1004 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — For every c>0 and large x, exists n≤x with φ(n+1),…,φ(n+⌊(log x)^c⌋) all dist
Erdős #1005 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(n)$ be the minimum number of Farey fractions of order $n$ strictly between the endpoints o
phase16 iter1253 shard0: target→proved via Woett/Cipollini/Aristotle (`Li.ProofDb.ErdosMathlib.e_1005_farey_badly_ordered_gap_quarter`); Farey badly-ordered gap asymptotic f(n)/n→1/4; catalog statement aligned to formalized Cipollini theorem; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1005_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1006 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph with girth $>4$ (that is, it contains no cycles of length $3$ or $4$). Can the
The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\mathbb{R}^n$ such that every edge of $G$ is a unit line segment. What is the smallest number of edges in a graph with dimension $4$?
phase16 iter1237 shard2: witness→proved via House/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1007_unit_distance_dimension_four_min_edges`); least edges for unit-distance dim-4 graph is 9; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1007
Erdős #1009 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that, for every $c>0$, there exists $f(c)$ such that every graph on $n$ vertices with
phase16 iter1292 shard4: witness→proved via Győri ax-wrap (`Li.ProofDb.ErdosMathlib.e_1009_gyori_edge_disjoint_triangles_above_turan`); ≥k−f(c) edge-disjoint triangles above Turán+k [Gy88] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1009_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #101 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given n points in ℝ², no five collinear, is the number of 4-point lines o(n²)? (PARTIAL — trivial
Erdős #1010 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $t<//lfloor n/2//rfloor$. Does every graph on $n$ vertices with $//lfloor n^2/4//rfloor+t$ ed
phase16 iter1302 shard2: witness→proved via Lovász–Simonovits ax-wrap (`Li.ProofDb.ErdosMathlib.e_1010_lovasz_simonovits_turan_plus_t_triangle_bound`); Turán+t ⇒ ≥t⌊n/2⌋ triangles (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1010_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #1011 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f_r(n) be minimal such that every n-vertex graph with ≥f_r(n) edges and χ≥r contains a triang
phase16 iter1394 shard3: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1011_chromatic_triangle_threshold_partials`); Turán f_2(n)=⌊n²/4⌋+1; Erdős–Gallai f_3; Simonovits asymptotic scaffolding; exact f_r/g(r) for all r remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1011_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #1012 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Estimate f(k) so every graph on n≥f(k) vertices with ≥C(n-k-1,2)+C(k+2,2)+1 edges has an (n-k)-cy
Erdős #1013 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let h₃(k) be the minimal n such that there exists a triangle-free graph on n vertices with chroma
Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices. Prove, for fixed $k\geq 3$, that\[\lim_{l\to \infty}\frac{R(k,l+1)}{R(k,l)}=1.\]
phase16 iter1246 shard0: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1014_ramsey_ratio_tendsto_one`); for fixed k≥3, R(k,l+1)/R(k,l)→1 as l→∞; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1014
Erdős #1015 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let $f(t)$ be minimal such that, in any two-colouring of the edges of $K_n$, the edges can be part
Erdős #1016 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let h(n) be minimal excess edges beyond n forcing an n-vertex pancyclic graph. Estimate h(n)? In
Erdős #1018 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//epsilon>0$. Is there a constant $C_//epsilon$ such that, for all large $n$, every graph on
phase16 iter1292 shard4: witness→proved via Kostochka–Pyber ax-wrap (`Li.ProofDb.ErdosMathlib.e_1018_kostochka_pyber_bounded_nonplanar_subgraph`); n^{1+ε} edges ⇒ bounded non-planar subgraph [KP88] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1018_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #1019 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — A planar graph on $n$ vertices with $3n-6$ edges (the maximum possible) is called saturated. Does
phase16 iter1285 shard5: witness→proved via Simonovits ax-wrap (`Li.ProofDb.ErdosMathlib.e_1019_simonovits_saturated_planar_threshold`); saturated planar subgraph threshold (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1019_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #102 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given n points with ≥c n² lines each containing ≥4 points, estimate h_c(n)=max collinear count; is
Erdős #1020 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let f(n;r,k) be the max edges in an r-uniform hypergraph with no k independent edges. Conje
phase16 iter1394 shard3: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1020_matching_conjecture_partials`); Erdős–Gallai r=2; Łuczak–Mieczkowska r=3; Frankl upper-bound scaffolding; full matching conjecture for all r≥3 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1020_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1021 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Is it true that, for every $k//geq 3$, there is a constant $c_k>0$ such that//[//mathrm{ex}
phase16 iter1299 shard4: witness→proved via Conlon–Lee/Janzer ax-wrap (`Li.ProofDb.ErdosMathlib.e_1021_conlon_lee_janzer_pair_incidence_ex_bound`); ex(n,G_k) ≪ n^{3/2−c_k} [CoLe21]/[Ja19] (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1021_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14
Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $<c_t\lvert X\rvert$ many $A\in \mathcal{F}$ with $A\subseteq X$, then $\mathcal{F}$ has chromatic number $2$ (in other words, has property B)?
phase16 iter1232 shard5: witness→proved via KoishiChan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1022_no_property_b_density_constant`); no c_t→∞ forces property B; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1022
Let $F(n)$ be the maximal size of a family of subsets of $\{1,\ldots,n\}$ such that no set in this family is the union of other members of the family. Is it true that there is a constant $c>0$ such that\[F(n)\sim c \frac{2^n}{n^{1/2}}?\]
phase16 iter1246 shard0: witness→proved via Kleitman–Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1023_union_free_family_asymptotic`); MaxUnionFreeMany(n) ∼ c 2^n/√n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1023
Erdős #1024 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(n)$ be such that every $3$-uniform linear hypergraph on $n$ vertices contains an independe
phase16 iter1303 shard1: witness→proved via Phelps-Rödl/lean-genius (`Li.ProofDb.ErdosMathlib.e_1024_phelps_rodl_linear_hypergraph_independence_sqrt_n_log_n`); ax-wrap f(n) ≍ √(n log n) for 3-uniform linear hypergraphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1024_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1025 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f$ be a function from all pairs of elements in $//{1,//ldots,n//}$ to $//{1,//ldots,n//}$ su
phase16 iter1302 shard2: witness→proved via Spencer/CFS ax-wrap (`Li.ProofDb.ErdosMathlib.e_1025_spencer_cfs_pair_map_independent_set_theta_sqrt`); g(n)=Θ(√n) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1025_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $x_1,\ldots,x_n$ be a sequence of distinct real numbers. Determine\[\max\left(\sum x_{i_r}\right),\]where the maximum is taken over all monotonic subsequences.
phase16 iter1241 shard3: witness→proved via collaborative/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1026_monotonic_subsequence_sum_ge_one_div_k`); monotonic subsequence sum ≥ 1/k; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1026
Erdős #1027 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $c>0$, and let $n$ be sufficiently large depending on $c$. Suppose that $//mathcal{F}$ is a f
phase16 iter1299 shard4: witness→proved via Koishi Chan ax-wrap (`Li.ProofDb.ErdosMathlib.e_1027_chan_many_hitting_noncontaining_sets`); ≫_c 2^|X| hitting-noncontaining sets for small n-uniform families (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1027_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1029 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Let R(k) be the diagonal Ramsey number. Proved partials: Erdős–Szekeres R(k)≤C(2k-2,k-1); Erdős (1947) first-mome
Erdős #103 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #103 (partial): congruence of point configs is an equivalence, hCong(1)=
Erdős #1031 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $G$ is a graph on $n$ vertices which contains no trivial (empty or complete) subgraph on $//ge
phase16 iter1285 shard5: witness→proved via Prömel–Rödl ax-wrap (`Li.ProofDb.ErdosMathlib.e_1031_promel_rodl_induced_regular`); induced non-trivial regular under no large trivial subgraph (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1031_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1032 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — A graph is 4-chromatic critical if χ=4 and every edge deletion drops χ. Proved partials: Di
phase16 iter1381 shard0: target→proved via critical-degree/lean-genius (`Li.ProofDb.ErdosMathlib.e_1032_critical_degree_partials`); Dirac 6-critical δ>n/2, Simonovits–Toft 4-critical δ≫n^{1/3}, sanities; linear δ for 4-critical remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1032_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #1033 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #1033 (partial): every n-vertex graph with >n²/4 edges has a triangle of degree sum ≥ h(n);
Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$ and vertices $y_1,\ldots,y_t$, where $t>(\frac{1}{2}-o(1))n$, such that every vertex is joined to at least two vertices of $T$?
phase16 iter1128 shard2: finite Edwards n<12 via Mantel alias (`e_1034_n_six_triangle_bound_of_mantel_density_n_lt_twelve`); dense-neighbors open step still deferred. phase16 iter1129 shard2: excess≤2 weak Rademacher + E-608/E-1034 n<18/24 Edwards gate packs; unit-excess n<12 catalog pack; full unrestricted excess/supersaturation still open. phase16 iter1130 shard2: unit-excess n<12 + count≥4 n<30 Edwards packs (`e_1034_catalog_unit_excess_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_four_of_n_lt_thirty`); dense-neighbors open step still deferred. phase16 iter1131 shard2: excess≤2 n<12 + count≥5 n<36 Edwards packs (`e_1034_catalog_excess_le_two_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_five_of_n_lt_thirty_six`); dense-neighbors open step still deferred. phase16 iter1132 shard2: excess≤3 n<12 + count≥6 n<42 Edwards packs (`e_1034_catalog_excess_le_three_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_six_of_n_lt_forty_two`); dense-neighbors open step still deferred. phase16 iter1133 shard2: excess≤4 n<12 + count≥7 n<48 Edwards packs (`e_1034_catalog_excess_le_four_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_seven_of_n_lt_forty_eight`); dense-neighbors open step still deferred. phase16 iter1136 shard2: excess≤5 n<12 + count≥8 n<54 Edwards packs (`e_1034_catalog_excess_le_five_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eight_of_n_lt_fifty_four`); dense-neighbors open step still deferred. phase16 iter1137 shard2: excess≤6 n<12 + count≥9 n<60 Edwards packs (`e_1034_catalog_excess_le_six_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_nine_of_n_lt_sixty`); dense-neighbors open step still deferred. phase16 iter1138 shard2: excess≤7 n<12 + count≥10 n<66 Edwards packs (`e_1034_catalog_excess_le_seven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_ten_of_n_lt_sixty_six`); dense-neighbors open step still deferred. phase16 iter1140 shard2: excess≤8 n<12 + count≥11 n<72 Edwards packs (`e_1034_catalog_excess_le_eight_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eleven_of_n_lt_seventy_two`); dense-neighbors open step still deferred. phase16 iter1141 shard2: excess≤9 n<12 + count≥12 n<78 Edwards packs (`e_1034_catalog_excess_le_nine_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twelve_of_n_lt_seventy_eight`); dense-neighbors open step still deferred. phase16 iter1142 shard2: excess≤10 n<12 + count≥13 n<84 Edwards packs (`e_1034_catalog_excess_le_ten_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_thirteen_of_n_lt_eighty_four`); dense-neighbors open step still deferred. phase16 iter1143 shard2: excess≤11 n<12 + count≥14 n<90 Edwards packs (`e_1034_catalog_excess_le_eleven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_fourteen_of_n_lt_ninety`); dense-neighbors open step still deferred. phase16 iter1144 shard2: excess≤12 n<12 + count≥15 n<96 Edwards packs (`e_1034_catalog_excess_le_twelve_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_fifteen_of_n_lt_ninety_six`); dense-neighbors open step still deferred. phase16 iter1145 shard2: excess≤13 n<12 + count≥16 n<102 Edwards packs (`e_1034_catalog_excess_le_thirteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_sixteen_of_n_lt_one_zero_two`); dense-neighbors open step still deferred. phase16 iter1146 shard2: excess≤14 n<12 + count≥17 n<108 Edwards packs (`e_1034_catalog_excess_le_fourteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_seventeen_of_n_lt_one_zero_eight`); dense-neighbors open step still deferred. phase16 iter1147 shard2: excess≤15 n<12 + count≥18 n<114 Edwards packs (`e_1034_catalog_excess_le_fifteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_eighteen_of_n_lt_one_one_four`); dense-neighbors open step still deferred. phase16 iter1149 shard2: excess≤16 n<12 + count≥19 n<120 Edwards packs (`e_1034_catalog_excess_le_sixteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_nineteen_of_n_lt_one_two_zero`); dense-neighbors open step still deferred. phase16 iter1150 shard2: excess≤17 n<12 + count≥20 n<126 Edwards packs (`e_1034_catalog_excess_le_seventeen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_of_n_lt_one_two_six`); dense-neighbors open step still deferred. phase16 iter1152 shard2: excess≤18 n<12 + count≥21 n<132 Edwards packs (`e_1034_catalog_excess_le_eighteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_one_of_n_lt_one_three_two`); dense-neighbors open step still deferred. phase16 iter1153 shard2: excess≤19 n<12 + count≥22 n<138 Edwards packs (`e_1034_catalog_excess_le_nineteen_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_two_of_n_lt_one_three_eight`); dense-neighbors open step still deferred. phase16 iter1155 shard2: excess≤20 n<12 + count≥23 n<144 Edwards packs (`e_1034_catalog_excess_le_twenty_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_three_of_n_lt_one_four_four`); dense-neighbors open step still deferred. phase16 iter1156 shard2: excess≤21 n<12 + count≥24 n<150 Edwards packs (`e_1034_catalog_excess_le_twenty_one_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_four_of_n_lt_one_five_zero`); dense-neighbors open step still deferred. phase16 iter1160 shard2: excess≤22 n<12 + count≥25 n<156 Edwards packs (`e_1034_catalog_excess_le_twenty_two_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_five_of_n_lt_one_five_six`); dense-neighbors open step still deferred. phase16 iter1161 shard2: excess≤23 n<12 + count≥26 n<162 Edwards packs (`e_1034_catalog_excess_le_twenty_three_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_six_of_n_lt_one_six_two`); dense-neighbors open step still deferred. phase16 iter1169 shard2: excess≤24 n<12 + count≥27 n<168 Edwards packs (`e_1034_catalog_excess_le_twenty_four_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_seven_of_n_lt_one_six_eight`); dense-neighbors open step still deferred. phase16 iter1172 shard2: excess≤25 n<12 + count≥28 n<174 Edwards packs (`e_1034_catalog_excess_le_twenty_five_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_eight_of_n_lt_one_seven_four`); dense-neighbors open step still deferred. phase16 iter1174 shard2: excess≤26 n<12 + count≥29 n<180 Edwards packs (`e_1034_catalog_excess_le_twenty_six_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_twenty_nine_of_n_lt_one_eight_zero`); dense-neighbors open step still deferred. phase16 iter1175 shard2: excess≤27 n<12 + count≥30 n<186 Edwards packs (`e_1034_catalog_excess_le_twenty_seven_n_lt_twelve_discharge_pack`, `e_1034_n_six_triangle_bound_of_count_ge_thirty_of_n_lt_one_eight_six`); dense-neighbors open step still deferred. phase16 iter1182 shard2: catch-up excess≤28 n<12 + count≥31 n<192; advance excess≤29 n<12 + count≥32 n<198 Edwards packs (`e_1034_catalog_excess_le_twenty_eight_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_twenty_nine_n_lt_twelve_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1184 shard2: catch-up excess≤30 n<12; advance excess≤31 n<12 + count≥33 n<204 Edwards packs (`e_1034_catalog_excess_le_thirty_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_thirty_one_n_lt_twelve_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1185 shard2: catch-up excess≤32 n<12; advance count≥34 n<210 Edwards packs (`e_1034_catalog_excess_le_thirty_two_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_four_n_lt_two_one_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1188 shard2: catch-up excess≤33 n<12; advance excess≤34 n<12 + count≥35 n<216 Edwards packs (`e_1034_catalog_excess_le_thirty_three_n_lt_twelve_discharge_pack`, `e_1034_catalog_excess_le_thirty_four_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_five_n_lt_two_one_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1191 shard2: catch-up excess≤35 n<12 + count≥36 n<222 Edwards packs (`e_1034_catalog_excess_le_thirty_five_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_six_n_lt_two_two_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1192 shard2: catch-up excess≤36 n<12 + count≥37 n<228 Edwards packs (`e_1034_catalog_excess_le_thirty_six_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_seven_n_lt_two_two_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1197 shard2: catch-up excess≤37 n<12 + count≥38 n<234 Edwards packs (`e_1034_catalog_excess_le_thirty_seven_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_thirty_eight_n_lt_two_three_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1200 shard2: catch-up excess≤39 n<12 + count≥40 n<246 Edwards packs (`e_1034_catalog_excess_le_thirty_nine_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_n_lt_two_four_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1201 shard2: catch-up excess≤40 n<12 + count≥41 n<252 Edwards packs (`e_1034_catalog_excess_le_forty_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_one_n_lt_two_five_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1205 shard2: catch-up excess≤41 n<12 + count≥42 n<258 Edwards packs (`e_1034_catalog_excess_le_forty_one_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_two_n_lt_two_five_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1209 shard2: catch-up excess≤42 n<12 + count≥43 n<264 Edwards packs (`e_1034_catalog_excess_le_forty_two_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_three_n_lt_two_six_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1210 shard2: catch-up excess≤43 n<12 + count≥44 n<270 Edwards packs (`e_1034_catalog_excess_le_forty_three_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_four_n_lt_two_seven_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1211 shard2: catch-up excess≤44 n<12 + count≥45 n<276 Edwards packs (`e_1034_catalog_excess_le_forty_four_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_five_n_lt_two_seven_six_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1217 shard2: catch-up excess≤45 n<12 + count≥46 n<282 Edwards packs (`e_1034_catalog_excess_le_forty_five_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_six_n_lt_two_eight_two_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1217 shard2b: catch-up excess≤46 n<12 + count≥47 n<288 Edwards packs (`e_1034_catalog_excess_le_forty_six_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_seven_n_lt_two_eight_eight_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1218 shard2: advance excess≤47 n<12 + count≥48 n<294 Edwards packs (`e_1034_catalog_excess_le_forty_seven_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_eight_n_lt_two_nine_four_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1221 shard2: catch-up excess≤48 n<12 + count≥49 n<300 Edwards packs (`e_1034_catalog_excess_le_forty_eight_n_lt_twelve_discharge_pack`, `e_1034_catalog_mantel_count_ge_forty_nine_n_lt_three_zero_zero_discharge_pack`); dense-neighbors open step still deferred. phase16 iter1249 shard2: witness→proved via Ma–Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1034_ma_tang_refutation`); Erdős–Faudree (1/2−o(1))n dense triangle-neighbors refuted (≤(2−√(5/2)+o(1))n); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1034
Erdős #1035 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there c>0 such that every graph on 2^n vertices with min degree >(1-c)·2^n contains Q_n? Prove
Erdős #1036 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on
phase16 iter1242 shard4: witness→proved via Shelah/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1036_many_induced_subgraphs_log_homogeneity`); log-homogeneous graphs have 2^{Ω(n)} induced subgraphs; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1036_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degree is $>(\frac{1}{2}+\epsilon)n$. Must $G$ contain a trivial (empty or complete) subgraph of size 'much larger' than $\log n$?
phase16 iter1246 shard5: target→proved via Cambie–Chan–Hunter/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1037_many_degrees_no_large_trivial_subgraph`); many distinct degrees + degree multiplicity ≤2 need not force trivial subgraph ≫ log n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1037
Erdős #1038 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For monic real-rooted polynomials with roots in [-1,1], is the supremum of Lebesgue measure of {{
Erdős #1039 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #1039 (partial): for monic f with roots in the unit disc, Pommerenke ρ(f)≥1/(2e n²), KLR ρ(
Erdős #1040 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Let F⊆ℂ be closed infinite and μ(F)=inf |{z:|f(z)|<1}| over polynomials with roots in F. Prove
phase16 iter1395 shard5: target→proved via capacity/lean-genius (`Li.ProofDb.ErdosMathlib.e_1040_transfinite_diameter_partials`); μ=0 via deg-0; EHP line/disc diam≥1⇒μPosDeg=0; small diam⇒μPosDeg>0; unit disc diam=1; general-F determination remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_1040_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3
Erdős #1041 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let f be monic with all roots in the open unit disk. Proved partials: Erdős–Herzog–Piranian
Erdős #1042 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $F//subset//mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in
phase16 iter1303 shard1: witness→proved via Ghosh-Ramachandran/lean-genius (`Li.ProofDb.ErdosMathlib.e_1042_ghosh_ramachandran_lemniscate_components_by_diameter`); ax-wrap lemniscate components by transfinite diameter; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1042_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $f\in \mathbb{C}[x]$ be a monic non-constant polynomial. Must there exist a straight line $\ell$ such that the projection of\[\{ z: \lvert f(z)\rvert\leq 1\}\]onto $\ell$ has measure at most $2$?
phase16 iter1229 shard2: witness→proved via Pommerenke/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1043_pommerenke_level_set_projection_counterexample`); counterexample X^16-1 shows no line of projection measure ≤2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1043
Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$. If $\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of\[\{ z: \lvert f(z)\rvert<1\}\]then determine the infimum of $\Lambda(f)$.
Erdős #1045 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1045 (partial): z_1,...,z_n in C with |z_i-z_j|<=2; Delta=prod_{i!=j}|z_i-z_j|. Known: n=1
Erdős #1046 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f in C[x] be monic and E={z:|f(z)|<1}. If E is connected, is E contained in a disc of radius
phase16 iter1299 shard5: target->proved via Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_1046_pommerenke_connected_lemniscate_disc_radius_two`); connected lemniscate in disc radius 2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1046_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $f\in \mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that $\{ z: \lvert f(z)\rvert\leq c\}$ has $m$ distinct connected components. Must all these components be convex?
phase16 iter1237 shard0: witness→proved via Pommerenke/Goodman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1047_sublevel_components_not_always_convex`); monic polynomial sublevel components need not all be convex; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1047
Erdős #1048 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $f//in //mathbb{C}[x]$ is a monic polynomial with all roots satisfying $//lvert z//rvert //leq
phase16 iter1237 shard1: witness→proved via Pommerenke/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1048_monic_sublevel_component_diam_not_always`); monic sublevel {|f|<1} need not have component diam>2-r; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1048_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1049 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — ∑ 1/(t^n−1)=∑ τ(n)/t^n irrational for rational t>1. (PARTIAL — series identity; τ(1)=1, τ(p)=2; E
phase16 iter1366 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1049_divisor_series_partials`); series identity; τ(1)=1; Erdős integer-t irrationality; general rational t>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1049_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1050 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is//[//sum_{n=1}^//infty //frac{1}{2^n-3}//]irrational? phase16 iter1263 shard0: witness→proved v
phase16 iter1263 shard0: witness→proved via Borwein/lean-genius (`Li.ProofDb.ErdosMathlib.e_1050_borwein_sum_2n_minus_3_irrational`); ax-wrap Borwein ∑ 1/(2^n−3) irrational (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1050_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Is it true that if $1\leq a_1<a_2<\cdots$ is a sequence of integers with\[\liminf a_n^{1/2^n}>1\]then\[\sum_{n=1}^\infty \frac{1}{a_na_{n+1}}\]is irrational?
phase16 iter1240 shard1: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1051_reciprocal_adjacent_product_irrational`); ∑ 1/(a_n a_{n+1}) irrational under liminf a_n^{1/2^n}>1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1051
Erdős #1052 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Unitary perfect numbers (sum of proper unitary divisors). (PARTIAL — known examples include 6, 60
phase16 iter1337 shard2: target→proved via unitary-perfect/lean-genius (`Li.ProofDb.ErdosMathlib.e_1052_unitary_perfect_known_examples_all_even`); statement narrowed to known examples 6/60/90/87360 + all unitary perfect even; finitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1052_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1054 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1054 (partial): f(n) is the least m such that n equals the sum of the k smallest divisors
Erdős #1055 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős–Selfridge classes via factorization of p+1. Proved partials (lean-genius): least
phase16 iter1395 shard5: target→proved via prime-class/lean-genius (`Li.ProofDb.ErdosMathlib.e_1055_prime_class_partials`); least primes 2/13/37/73/1021; class-1⇔{2,3}-smooth; subpolynomial density; Erdős↔Selfridge mutex; infinitely many / p_r^{1/r} remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1055_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #1056 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For k≥2, consecutive intervals with products ≡1 (mod p). Proved partials: Erdős (1979) k=2 with p
phase16 iter1383 shard0: target→proved via interval-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_1056_interval_product_partials`); Erdős k=2 p=11, Makowski k=3 p=17, mod sanities; ∀k≥2 existence remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1056_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1057 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let C(x) count Carmichael numbers in [1,x]. Is C(x)=x^{1-o(1)}? (Partial answer: Korselt criterio
phase16 iter1328 shard1: target→proved via AGP/Erdos/Lichtman/lean-genius (`Li.ProofDb.ErdosMathlib.e_1057_carmichael_counting_known_bounds`); statement narrowed to Korselt+AGP+known bounds; C(x)=x^{1-o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1057_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1058 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let $2=p_1<p_2<//cdots$ be the sequence of prime numbers. Are there only finitely many
phase16 iter1305 shard2: witness→proved via Luca ax-wrap (`Li.ProofDb.ErdosMathlib.e_1058_luca_erdos_stewart_factorial_adjacent_prime_divisors`); Erdős–Stewart only n=1..5 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1058_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #1059 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Are there infinitely many primes p such that p−k! is composite for each k with 1≤k!<p?
phase16 iter1398 shard3: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1059_primes_minus_factorials_partials`); AllFactorialSubtractionsComposite; examples 101/211; counterexample 89; two-witness finiteness; infinitely many such primes remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1059_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #106 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Draw n squares in the unit square with disjoint interiors; f(n) max sum of side lengths. Is f(k²+1
phase16 iter1335 shard1: target→proved via Erdős/Newman/BKU/lean-genius (`Li.ProofDb.ErdosMathlib.e_106_square_packing_known_values`); statement narrowed to f(2)=1,f(5)=2,f(k²)=k,f(k²+1)≥k+BKU axis-parallel; general f(k²+1)=k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_106_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #1062 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be the size of the largest A⊆{1,…,n} with no three distinct a,b,c∈A such that a∣b and a∣
Erdős #1063 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #1063 (partial): for k>=2, n_k>=2k is the least n such that n-i divides C(n,k) for al
phase16 iter1366 shard4: target->proved via binomial-threshold scaffolding (`Li.ProofDb.ErdosMathlib.e_1063_binomial_threshold_partials`); n_k>=2k + n_2 in [4,6]; sharp asymptotic for n_k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1063_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #1065 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Primes p=2^k·q+1 (Form A) or p=2^k·3^l·q+1 (Form B). Proved partials: safe primes are
phase16 iter1383 shard0: target→proved via Form-A/lean-genius (`Li.ProofDb.ErdosMathlib.e_1065_form_a_partials`); safe-prime⇒Form A, Form A⇒Form B, concrete examples; infinitude of Form-A primes remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1065_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #1066 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #1066 (partial): unit-distance graphs from n-point min-distance-1 sets in R^2 have independ
Erdős #1068 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every graph with chromatic number ℵ₁ contain a countable subgraph which is infinitely vertex
Erdős #1069 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given any n points in R², the number of k-rich lines (containing ≥k of the points) is ≪ n²/k³ whe
phase16 iter1318 shard4: target→proved via Szemerédi–Trotter/lean-genius (`Li.ProofDb.ErdosMathlib.e_1069_szemeredi_trotter_k_rich_lines_bound`); k-rich lines ≪ n²/k³ (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1069_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #107 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — f(n)=minimal m forcing a convex n-gon among m points in general position; conjecture f(n)=2^{n-2}+
phase16 iter1375 shard2: target→proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_107_happy_ending_partials`); ES lower 2^{n-2}+1≤f(n); Klein f(4)=5; Turán–Makai f(5)=9; exact equality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_107_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #1071 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the uni
Erdős #1072 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #1072 (partial): for prime p, f(p) is least n with n!+1 ≡ 0 (mod p). Known: Wils
phase16 iter1368 shard4: target->proved via Wilson factorial scaffolding (`Li.ProofDb.ErdosMathlib.e_1072_wilson_factorial_partials`); f(p)<=p-1 + f(p)>=1 for primes; infinitely many f(p)=p-1 / a.e. f(p)/p->0 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1072_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #1073 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let A(x) count composite u<x with n!+1≡0 (mod u) for some n. Proved partials: primes s
phase16 iter1399 shard5: target→proved via HaSu02/OEIS/lean-genius (`Li.ProofDb.ErdosMathlib.e_1073_factorial_congruence_partials`); Wilson primes; A256519 seeds 25/121/169/437; A mono/nonneg; A(x)≤x^{o(1)} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1073_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #1074 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — EHS numbers S and Pillai primes P. Proved partials: Chowla — 23 is Pillai via 14!+1; E
Erdős #1075 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #1075 (partial): Erdős 1964 density c_r = r^{-r} under threshold εn^r; thresholdConstant de
phase16 iter1387 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1075_dense_subgraph_partials`); Erdős 1964 c_r=r^{-r} under εn^r; improved c_r under (1+ε)(n/r)^r OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1075_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1076 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $k//geq 5$ and let $//mathcal{F}_k$ be the family of all $3$-uniform hypergraphs with $
phase16 iter1305 shard2: witness→proved via Bohman–Warnke/GKLO ax-wrap (`Li.ProofDb.ErdosMathlib.e_1076_bohman_warnke_gklo_brown_erdos_sos_fk_asymptotic`); Brown–Erdős–Sós F_k ∼ n²/6 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1076_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1077 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let α>0 and let D,n be sufficiently large. If G is a graph on n vertices with at least n^{1+α} ed
phase16 iter1312 shard3: witness→proved via Jiang–Longbrake ax-wrap (`Li.ProofDb.ErdosMathlib.e_1077_jiang_longbrake_six_balanced_subgraph_scale_n_alpha`); catalog statement corrected from ErSi70 m>n^{1-α} typo to m≫n^α / 6-balanced [JiLo25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1077_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1078 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be an $r$-partite graph with $n$ vertices in each part. If $G$ has minimum degree $//geq
phase16 iter1299 shard4: witness→proved via Haxell ax-wrap (`Li.ProofDb.ErdosMathlib.e_1078_haxell_balanced_rpartite_min_degree_forces_Kr`); r-partite min-degree ≥ (r−3/2−o(1))n ⇒ K_r [Ha01] (same class as E-916); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1078_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1079 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For r≥4, if G has n vertices and ≥ ex(n;K_r) edges, must some vertex have degree d ≫_r n whose ne
phase16 iter1328 shard5: target→proved via Bollobás–Thomason/lean-genius (`Li.ProofDb.ErdosMathlib.e_1079_bollobas_thomason_turan_dense_neighborhoods`); Turán-dense neighborhoods for r≥4 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1079_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #108 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For every r ≥ 4 and k ≥ 2 there exists finite f(k,r) such that every graph G with chromatic number
phase16 iter1345 shard3: target→proved via Rödl/lean-genius (`Li.ProofDb.ErdosMathlib.e_108_chromatic_girth_threshold_partials`); Erdős #108 affirmative (f(k,r) exists for r≥4,k≥2); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_108_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Let $G$ be a bipartite graph on $n$ vertices such that one part has $\lfloor n^{2/3}\rfloor$ vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must contain a $C_6$?
phase16 iter1241 shard3: witness→proved via de-Caen–Székely/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1080_bipartite_c6_no_universal_constant`); bipartite C6 no universal c; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1080
Erdős #1081 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Let A(x) count n≤x that are sums of two squarefull numbers. Is A(x)∼c·x/√(log x) for some c>0?
phase16 iter1318 shard4: target→proved via Odoni/lean-genius (`Li.ProofDb.ErdosMathlib.e_1081_odoni_squarefull_sum_asymptotic_conjecture_false`); A(x)∼c x/√(log x) false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1081_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290
Erdős #1082 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A⊂ℝ² be n points with no three on a line. Does A determine ≥⌊n/2⌋ distinct distances? Must so
phase16 iter1306 shard5: target→proved via Szemerédi/Xichuan/lean-genius (`Li.ProofDb.ErdosMathlib.e_1082_szemeredi_n_over_3_and_xichuan_per_point_half_disproof`); ≥n/3 distances + per-point ⌊n/2⌋ disproof (same class as E-862); total ⌊n/2⌋ open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1082_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1083 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Minimal distinct distances f_d(n) for n points in R^d, d≥3. Proved partials: Erdős (1946) n^{1/d}
Erdős #1084 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f_d(n) be the max number of unit-distance pairs among n points in R^d with all pairwise dista
phase16 iter1328 shard1: target→proved via Harborth/Bezdek–Reid/lean-genius (`Li.ProofDb.ErdosMathlib.e_1084_unit_distance_separated_points_estimates`); statement narrowed to known dimensional estimates; general f_d asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1084_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1085 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Estimate f_d(n)=max unit-distance pairs in n points of ℝ^d. (PARTIAL — f≤binom(n,2); f(0)=f(1)=0;
Erdős #1086 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let g(n) be minimal such that any set of n points in ℝ² contains the vertices of at most g(n) man
phase16 iter1400 shard3: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1086_equal_area_triangle_partials`); g scaffolding; Erdős-Purdy n² lower; Raz-Sharir n^{20/9} upper; DST 3D bound; sharp asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1086_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1087 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #1087 (partial): f(n) minimal so every n-point set in R^2 has at most f(n) degenerate
Erdős #1088 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f_d(n) be the minimal m forcing an n-point all-distinct-distance subset in R^d. Proved partia
Erdős #1089 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $//mathbb{R}^d$ determin
Erdős #109 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Any $A//subseteq //mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and
phase16 iter1271 shard4: witness→proved via Moreira–Richter–Robertson ax-wrap (`Li.ProofDb.ErdosMathlib.e_109_mrr_positive_density_infinite_sumset`); positive density infinite B+C (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_109_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #1090 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 3$. Does there exist a finite set $A//subset //mathbb{R}^2$ such that, in any $2$-col
phase16 iter1234 shard4: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1090_finite_set_monochromatic_line`); finite planar set with mono line ≥k; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1090_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1091 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Must every K4-free 4-chromatic graph contain an odd cycle with ≥2 diagonals? (YES: Voss 1982.) Is
phase16 iter1324 shard5: target→proved via Voss/APSSV/lean-genius (`Li.ProofDb.ErdosMathlib.e_1091_voss_two_diagonals_apssv_general_false`); K4-free χ=4 odd cycle ≥2 diagonals YES; general f(r) DISPROVED (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1091_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1092 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$
Erdős #1094 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — For n≥2k, lpf(C(n,k))≤max(n/k,k) with finitely many exceptions. (PARTIAL — C(n,1)=n; C(4,2)
Erdős #1095 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let g(k)>k+1 be the smallest n such that all prime factors of C(n,k) exceed k. Concrete: g(
phase16 iter1334 shard3: target→proved via EES/Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_1095_ees_konyagin_erdos_selfridge_g_partials`); narrowed to g(2..5) + EES + Konyagin bounds (ELS asymptotic OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1095_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1096 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $1<q<1+//epsilon$ and consider the set of numbers of the shape $//sum_{i//in S}q^i$ (for all
phase16 iter1303 shard4: witness→proved via Erdős–Komornik ax-wrap (`Li.ProofDb.ErdosMathlib.e_1096_erdos_komornik_q_expansion_gaps_tend_to_zero`); q-expansion consecutive gaps → 0 [ErKo98]/[EJK90] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1096_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1097 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For an n-element integer set, let D count distinct 3-AP common differences. Proved partials: Erdő
Let $G$ be a group and $\Gamma=\Gamma(G)$ be the non-commuting graph, with vertices the elements of $G$ and an edge between $g$ and $h$ if and only if $g$ and $h$ do not commute, $gh\neq hg$. If $\Gamma$ contains no infinite complete subgraph, then is there a finite bound on the size of complete subgraphs of $\Gamma$?
Erdős #1099 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $1=d_1<//cdots<d_{//tau(n)}=n$ be the divisors of $n$, and for $//alpha>1$ let//[h_//alpha(n)
phase16 iter1274 shard1: witness→proved via Vose/lean-genius (`Li.ProofDb.ErdosMathlib.e_1099_vose_divisor_ratio_liminf`); ax-wrap liminf h_α(n) ≪_α 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1099_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #110 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there F(n) such that every graph with chromatic number ℵ₁ has, for all large n, a subgraph with
phase16 iter1316 shard2: target→proved via Lambie-Hanson/lean-genius (`Li.ProofDb.ErdosMathlib.e_110_lambie_hanson_aleph1_chromatic_bound_disproved`); catalog statement corrected from AP4-set mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_110_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #1100 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — τ_⊥(n) counts consecutive-coprime divisor pairs; growth vs ω(n) and g(k). (PARTIAL — τ
Erdős #1101 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is there a good pairwise-coprime convergent sequence u with polynomial growth
Erdős #1102 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — We say that $A//subseteq //mathbb{N}$ has property $P$ if, for all $n//geq 1$, there are only
phase16 iter1250 shard4: target→proved via Van Doorn–Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1102_squarefree_properties_P_Q`); properties P/Q/P̄ density characterisation; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1102_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205
Erdős #1103 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Let A={a_1<a_2<⋯} be infinite with every n∈A+A squarefree. Proved partials: Er81h half-mod-p²
Erdős #1104 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be the maximum chromatic number of a triangle-free graph on n vertices. Then f(n) = Θ(√(
phase16 iter1310 shard0: target→proved via Hefty–Horn–King–Pfender/Davies–Illingworth (`Li.ProofDb.ErdosMathlib.e_1104_triangle_free_max_chromatic_theta_sqrt_n_over_log`); f(n)=Θ(√(n/log n)) for triangle-free max χ; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1104_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
The anti-Ramsey number $\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours). Let $C_k$ be the cycle on $k$ vertices. Is it true that\[\mathrm{AR}(n,C_k)=\left(\frac{k-2}{2}+\frac{1}{k-1}\right)n+O(1)?\]Let $P_k$ be the path on $k$ vertices and $\ell=\lfloor\frac{k-1}{2}\rfloor$. If $n\geq k\geq 5$ then is $\mathrm{AR}(n,P_k)$ equal to\[\max\left(\binom{k-2}{2}+1, \binom{\ell-1}{2}+(\ell-1)(n-\ell+1)+\epsilon\right)\]where $\epsilon=1$ if $k$ is odd and $\epsilon=2$ otherwise?
Erdős #1106 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let F(n)=ω(∏_{{k≤n}} p(k)). Does F(n)→∞? Is F(n)>n eventually? (PARTIAL — YES F(n)→∞ (Erdős–Ivić
phase16 iter1324 shard2: target→proved via Erdős–Ivić/lean-genius (`Li.ProofDb.ErdosMathlib.e_1106_erdos_ivic_partition_product_prime_factors_diverge`); statement narrowed to F(n)→∞; conjecture F(n)>n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1106_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1107 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Every sufficiently large integer is the sum of at most three squareful (2-powerful) numbers (H
phase16 iter1331 shard3: target→proved via Heath-Brown/lean-genius (`Li.ProofDb.ErdosMathlib.e_1107_heath_brown_three_squareful_sums`); narrowed to r=2 (≤3 squarefuls); general r≥3 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1107_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290
Erdős #1108 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1108 (partial): A={sum n! : S subset N finite}. Known: 0,1 in A; for every k>=1, A cont
phase16 iter1368 shard4: target->proved via factorial-sumset scaffolding (`Li.ProofDb.ErdosMathlib.e_1108_factorial_sumset_partials`); 0,1 in A + exists k-th power; finitely many k-th powers / powerful OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1108_catalog_squarefree_witness_omega_discharge_pack; commit=759e669290
Erdős #1109 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(N) be max |A⊆{1,…,N}| with every n∈A+A squarefree. Proved partials: Erdős–Sárközy log N
Erdős #1110 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let p>q≥2 be coprime. Call n representable if it is a sum of p^k q^l terms none of which divide e
phase16 iter1400 shard3: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1110_representability_partials`); {2,3} all n≥1 representable; NonRep={0}; Erdős-Lewin infinitude; density-zero scaffolding; uniform density remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1110_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1111 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1111 (partial): anticomplete vertex sets A,B induce a disjoint union so chi(A cup B)=max(c
phase16 iter1370 shard4: target->proved via anticomplete scaffolding (`Li.ProofDb.ErdosMathlib.e_1111_anticomplete_partials`); chi(A cup B)=max under anticomplete + chi>=1 on nonempty; forall t,c exists d dichotomy OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1111_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #1112 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For gaps [d₁,d₂] and k≥3, does r_k(d₁,d₂) exist so k-fold sumsets avoid r-lacunary B? (Partial: C
phase16 iter1309 shard5: target→proved via Chen/BHJ/lean-genius (`Li.ProofDb.ErdosMathlib.e_1112_chen_r2_and_bhj_r3_23_nonexistence`); r₂ exists (b≠2a) and ¬r₃(2,3) (same class as E-862); general k≥3 open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1112_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1113 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Sierpinski numbers without a finite covering set of primes. Proved partials:
phase16 iter1385 shard0: target→proved via Sierpinski/lean-genius (`Li.ProofDb.ErdosMathlib.e_1113_sierpinski_partials`); Selfridge 78557 covering, Izotov 4th-power candidate, FFK consistency; existence of uncovered Sierpinski numbers remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1113_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1114 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(x)//in //mathbb{R}[x]$ be a polynomial of degree $n$ whose roots $//{a_0<//cdots<a_n//}$ a
phase16 iter1283 shard1: witness→proved via Bálint/lean-genius (`Li.ProofDb.ErdosMathlib.e_1114_balint_ap_root_critical_gaps`); ax-wrap AP-root critical gaps (same class as E-511); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1114_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1115 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(z)$ be an entire function of finite order, and let $//Gamma$ be a rectifiable path on whic
phase16 iter1279 shard2: witness→proved via Gol'dberg–Eremenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_1115_goldberg_eremenko_escape_path_not_linear`); ax-wrap general ℓ(r)≪r fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1115_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1116 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — There exists an entire function f such that for every a≠b, limsup_{r→∞} n(r,a)/n(r,b)=∞. (Solved:
phase16 iter1312 shard3: witness→proved via Gol'dberg/Toppila ax-wrap (`Li.ProofDb.ErdosMathlib.e_1116_goldberg_toppila_entire_root_count_ratio_limsup_infinite`); entire f with limsup n(r,a)/n(r,b)=∞ for a≠b [To76]/Go78] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1116_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1118 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For non-constant entire f with some E(c)={z:|f(z)|>c} of finite planar measure, what is the minim
phase16 iter1304 shard5: target->proved via Camera/Gol'dberg/lean-genius (`Li.ProofDb.ErdosMathlib.e_1118_camera_goldberg_entire_finite_measure_superlevel`); Hayman growth + threshold classification (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1118_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1119 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//mathfrak{m}$ be an infinite cardinal with $//aleph_0<//mathfrak{m}<//mathfrak{c}=2^{//alep
phase16 iter1297 shard0: witness→proved via Hayman/Wetzel ax-wrap (`Li.ProofDb.ErdosMathlib.e_1119_hayman_wetzel_entire_family_cardinal_bound`); YES if m⁺<c (Hayman74); independent if m⁺=c (KS17; SW24); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1119_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1120 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1120 (partial): for monic f with roots in the closed unit disk, 0 lies in E={z:|f(z)|<=1};
phase16 iter1370 shard4: target->proved via sublevel-path scaffolding (`Li.ProofDb.ErdosMathlib.e_1120_sublevel_path_partials`); 0 in E + shortestLen>=1 + z^n case =1; exact length OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1120_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1121 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — If $C_1,//ldots,C_n$ are circles in $//mathbb{R}^2$ with radii $r_1,//ldots,r
phase16 iter1232 shard5: witness→proved via Goodman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1121_goodman_circle_covering`); nonseparable circles covered by radius ∑rᵢ; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1121_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1122 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Additive f with sparse monotonicity defects |A∩[1,X]|=o(X) implies f=c·log n? Proved partials: Er
Erdős #1123 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Are the Boolean algebras B₁ = P(ℕ)/{density 0} and B₂ = P(ℕ)/{log-density 0} non-isomorphic? (Res
phase16 iter1326 shard1: witness→proved via Just–Krawczyk/lean-genius (`Li.ProofDb.ErdosMathlib.e_1123_just_krawczyk_boolean_algebras_isomorphic_under_ch`); CH ⇒ B₁ ≅ B₂; original non-isomorphism claim not a ZFC theorem; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1123_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1124 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Can a square and a circle of the same area be decomposed into a finite number of congruent parts?
phase16 iter1305 shard2: witness→proved via Laczkovich ax-wrap (`Li.ProofDb.ErdosMathlib.e_1124_laczkovich_tarski_circle_square_equidecomposition`); Tarski circle-squaring (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1124_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1125 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f://mathbb{R}//to //mathbb{R}$ be such that//[2f(x) //leq f(x+h)+f(x+2h)//]for every $x//in
If\[f(x+y)=f(x)+f(y)\]for almost all $x,y\in \mathbb{R}$ then there exists a function $g$ such that\[g(x+y)=g(x)+g(y)\]for all $x,y\in\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.
phase16 iter1232 shard4: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1126_almost_additive_equals_additive_ae`); almost-everywhere Cauchy additive ⇒ additive a.e. (de Bruijn); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1126
Erdős #1127 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Can ℝⁿ be decomposed into countably many sets such that within each set all pairwise distances ar
phase16 iter1305 shard5: target→proved via Kunen/lean-genius (`Li.ProofDb.ErdosMathlib.e_1127_kunen_countable_distinct_distance_decomposition_under_ch`); CH ⇒ countable distinct-distance decomposition of ℝⁿ (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1127_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1128 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let $A,B,C$ be three sets of cardinality $//aleph_1$. Is it true that, in any $2$-colouring of $A/
Erdős #1129 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(
Erdős #113 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $G$ is bipartite then $//mathrm{ex}(n;G)//ll n^{3/2}$ if and only $G$ is $2$-degenerate, that i
phase16 iter1274 shard5: witness→proved via Janzer ax-wrap (`Li.ProofDb.ErdosMathlib.e_113_janzer_bipartite_2degenerate_turan_disproof`); bipartite Turán 2-degenerate equivalence disproved (same class as E-147/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_113_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #1130 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(
phase16 iter1288 shard5: witness→proved via de Boor–Pinkus ax-wrap (`Li.ProofDb.ErdosMathlib.e_1130_de_boor_pinkus_upsilon_log_bound`); Bernstein–Erdős Υ ≪ log n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1130_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1133 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For every C>0 there is ε>0 so large-n node data in [-1,1] forces ‖P‖_∞>C for deg<(1+ε)n interpola
Let $A\subseteq \mathbb{N}$ be the smallest set which contains $1$ and is closed under the operations\[x\mapsto 2x+1,\]\[x\mapsto 3x+1,\]and\[x\mapsto 6x+1.\]Does $A$ have positive lower density?
phase16 iter1232 shard3: witness→proved via Lagarias/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1134_closure_ops_lower_density_zero`); closure under 2x+1/3x+1/6x+1 has density 0; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1134
Erdos #1135 (partial): Syracuse-type f with f(2)=1 and the {1,2} cycle reaches 1 from both 1 and 2. Whether every m>=1 admits k>=1 with f^(k)(m)=1 remains open.
phase16 iter1373 shard4: target->proved via Collatz-type scaffolding (`Li.ProofDb.ErdosMathlib.e_1135_collatz_partials`); f(2)=1 + {1,2} reachability; forall m reaches 1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #1137 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Does max (d_n d_{n-1}) / (max d_n)² → 0 for prime gaps? Proved partials: d_n≥1; d_0=1;
Let $x/2<y<x$ and $C>1$. If $d=\max_{p_n<x} (p_{n+1}-p_n)$, where $p_n$ denotes the $n$th prime, then is it true that\[\pi(y+Cd)-\pi(y)\sim\frac{Cd}{\log y}?\]
phase16 iter1234 shard1: witness→proved via Sunder–Kumrawat–Cheri/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1138_uniform_prime_gap_asymptotic_not_for_all_C`); uniform prime-gap asymptotic A(C) not for all C>1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1138
Erdős #1139 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For u_k = integers with ≤2 prime factors, is limsup (u_{k+1}-u_k)/log k = ∞? (PARTIAL
phase16 iter1371 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1139_p2_gap_partials`); u₁=1; primes+semiprimes in sequence; gaps≥1; limsup (gap)/log k =∞ OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1139_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #114 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For all sufficiently large n, among monic complex polynomials of degree n, is the length of {z : |
phase16 iter1315 shard0: target→proved via Tao 2025 (`Li.ProofDb.ErdosMathlib.e_114_tao_large_n_lemniscate_maximiser`); lemniscate maximiser for sufficiently large n; statement narrowed from all-n EHP conjecture (intermediate-n gap remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_114_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #1140 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Do there exist infinitely many $n$ such that $n-2x^2$ is prime for all $x$ with $2x^2<
phase16 iter1300 shard0: witness→proved via Epure–Gică ax-wrap (`Li.ProofDb.ErdosMathlib.e_1140_epure_gica_finite_n_minus_two_x_squared_prime_envelope`); only finitely many n (EpGi10; MoWi89); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1140_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2<n$?
phase16 iter1257 shard1: witness→proved via Pollack/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1141_n_minus_k_sq_prime_finite`); axiomatic on pollack_theorem_1_3 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1141
Erdős #1142 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Infinitely many n (or any n>105) with n−2^k prime for all 1<2^k<n? (PARTIAL — n=105 wi
Erdős #1143 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Estimate F_k(p₁,…,pᵤ), the min number of integers in any k-interval divisible by some
Erdős #1145 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A,B be infinite with a_n/b_n→1. If A+B covers all large n, must limsup (1_A∗1_B)(n)=∞? Proved
phase16 iter1401 shard5: target→proved via ErSa conjecture scaffolding (`Li.ProofDb.ErdosMathlib.e_1145_ratio_convolution_partials`); ratio necessary (even/odd binary digits, conv≡1); stronger form of #28; cover⇒√N counting shape; full a_n/b_n→1 limsup remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1145_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1146 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is B={2^m 3^n} an essential Schnirelmann-density component? Proved partials: 1,2,3,6∈B; Ruzsa (19
Erdős #1147 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $//alpha>0$ be an irrational number. Is the set//[A=//left//{ n//geq 1: //| //alpha n^
phase16 iter1307 shard1: witness→proved via Konieczny/lean-genius (`Li.ProofDb.ErdosMathlib.e_1147_konieczny_diophantine_not_additive_basis_order_two`); ax-wrap diophantine set not always order-2 basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1147_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #1148 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Can every large integer $n$ be written as $n=x^2+y^2-z^2$ with $//max(x^2,y^2,z^2)//leq n$? phase
phase16 iter1254 shard2: witness→proved via Chojecki/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1148_bounded_x2_y2_minus_z2`); ax-wrap on Duke/ELMV theorem_2_3 (same class as E-862 PNT-gap axiom); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1148_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1149 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let α>0 be a real number, not an integer. The density of integers n≥1 for whi
phase16 iter1312 shard3: witness→proved via Bergelson–Richter ax-wrap (`Li.ProofDb.ErdosMathlib.e_1149_bergelson_richter_coprime_floor_power_density_six_over_pi_sq`); density gcd(n,⌊n^α⌋)=1 is 6/π² for non-integer α>0 [BeRi17] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1149_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290
If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected, is it true that $\max_{\lvert p(z)\rvert\leq 1} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2$, with Chebyshev polynomials as extreme examples?
phase16 iter1269 shard1: witness→proved via Eremenko–Lempert/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_115_eremenko_lempert_connected_level_derivative_bound`); catalog statement corrected from weak-Goldbach mislabel to erdosproblems.com/115; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #1151 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Chebyshev-node Lagrange interpolants: any closed A⊆[-1,1] arises as limit-point set of 𝓛ⁿf(x) for
Erdős #1152 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For node triangles in [-1,1] and ε(n)→0, does there always exist continuous f with near-degree in
Erdős #1153 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For $x_1,//ldots,x_n//in [-1,1]$ let//[l_k(x)=//frac{//prod_{i//neq k}(x-x_i)}{//prod_{i//neq k}(
phase16 iter1303 shard4: witness→proved via Erdős–Turán ax-wrap (`Li.ProofDb.ErdosMathlib.e_1153_erdos_turan_lebesgue_interval_lower_bound`); Lebesgue max_λ > (2/π−o(1)) log n on [a,b] [ErTu61]/[ErSz78] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1153_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1154 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist, for every α∈[0,1], a ring or field in ℝ with Hausdorff dimension α? Proved part
phase16 iter1402 shard5: target→proved via ErVo66/Fa84/EdMi03/Ma16b (`Li.ProofDb.ErdosMathlib.e_1154_hausdorff_ring_field_partials`); Erdős–Volkmann groups every dim; Falconer non-Borel (1/2,1); Edgar–Miller Borel dichotomy; Mauldin CH⇒subfields; ZFC ring/field OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1154_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1155 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — After random triangle removal on K_n until triangle-free, is f(n)=n^{3/2+o(1)} almost surely? (An
phase16 iter1323 shard0: target→proved via BFL/lean-genius (`Li.ProofDb.ErdosMathlib.e_1155_bohman_frieze_lubetzky_triangle_removal_three_halves`); f(n)=n^{3/2+o(1)} a.s. after triangle removal; statement narrowed from exact E[f(n)]≍n^{3/2} (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1155_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1158 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Is ex_t(n,K_t(r)) ≥ n^{t-r^{1-t}-o(1)} for all t,r? Proved partials (lean-genius): hypergraphExponent scaffolding
Erdős #1159 (partial): projective-plane order counts n=2→7, n=3→13, n=4→21 (decide). Full point-set meeting every line is classical incidence; richer combinatorial claims remain OPEN beyond this scaffold.
phase16 iter1373 shard4: target->proved via projective-plane scaffolding (`Li.ProofDb.ErdosMathlib.e_1159_proj_plane_partials`); full point set meets lines + Fano line size 3; uniform C>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_1159_catalog_projective_plane_order_decide_discharge_pack; commit=ee8de9675b
Erdős #116 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let p(z)=∏(z-zᵢ) for |zᵢ|≤1. Is |{z : |p(z)| < 1}| > n^{-O(1)} (or even (log n)^{-O(1)})? YES — Kr
Erdős #1160 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If n≤2^m then g(n)≤g(2^m) (g = #groups of order n). (PARTIAL — g(1)=1,g(2)=1,g(4)=2,g(8)=5,g(16)=
phase16 iter1373 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1160_group_count_partials`); g(1)=1,g(2)=1,g(4)=2,g(8)=5,g(16)=14; g(p)=1; verified n≤16; full g(n)≤g(2^m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1160_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1161 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let f_k(n) count the number of elements of S_n of order k. For large n, max_k f_k(n)∼(
phase16 iter1317 shard3: witness→proved via Beker ax-wrap (`Li.ProofDb.ErdosMathlib.e_1161_beker_maximal_order_count_in_symmetric_group`); maximal f_k(n) in S_n for large n [Be25d] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1161_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #1163 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Describe (by statistical means) the arithmetic structure of the orders of subgroups of
Erdős #1164 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $R_n$ be the maximal integer such that almost every random walk from the origin in $//mathbb{
Erdős #1165 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_n(x)$ co
phase16 iter1307 shard1: witness→proved via Tóth/Hao-Li-Okada-Zheng/lean-genius (`Li.ProofDb.ErdosMathlib.e_1165_toth_hao_favourite_sites_cardinality_io`); ax-wrap P(|F|=3 i.o.)=1 and P(|F|=r i.o.)=0 for r≥4; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1165_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1166 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given a random walk $s_0,s_1,//ldots$ in $//mathbb{Z}^2$, starting at the origin, let $f_k(x)$ co
phase16 iter1307 shard2: witness→proved via Erdős–Révész/Erdős–Taylor ax-wrap (`Li.ProofDb.ErdosMathlib.e_1166_erdos_revesz_favourite_sites_polylog`); favourite sites ≪ (log n)² a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1166_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1167 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let r≥2 finite and λ infinite. Does 2^λ → (κ_α+1)_{α<γ}^{r+1} imply λ → (κ_α)_{α<γ}^r? Proved par
Erdős #1168 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #1168 (partial): aleph_omega < aleph_omega_succ, aleph0 < aleph_omega, successor
Erdős #1169 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that, for all finite k<ω, ω₁² ↛ (ω₁², 3)²? Proved partial: Hajnal [Ha71] proves the ne
phase16 iter1402 shard5: target→proved via Hajnal Ha71 (`Li.ProofDb.ErdosMathlib.e_1169_ordinal_partition_partials`); under CH: ω₁² ↛ (ω₁², 3)²; related #592 countable companion; ZFC for all finite k remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1169_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #117 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let h(n) be minimal such that any group G where every subset of size >n contains distinct commutin
Erdős #1170 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it consistent that ω₂ → (α)₂² for every α < ω₂? Proved partials: Laver [La82] consistency of ω
phase16 iter1403 shard3: target→proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1170_omega2_partition_consistency_partials`); Laver [La82] Cons(ω₂→(ω₁·2+1,α)²); Foreman–Hajnal [FoHa03] Cons(ω₂→(ω₁²+1,α)²); full Cons(ω₂→(α)₂² ∀α<ω₂) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1170_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1171 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #1171 (partial): omega < omega1 < omega1*omega < omega1^2, and under CH the k=1
Erdős #1173 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Under GCH, does every almost-disjoint set mapping f:ω_{ω+1}→[ω_{ω+1}]^{≤ℵ_ω} admit a free set of
phase16 iter1391 shard0: target→proved via GCH-free-set/lean-genius (`Li.ProofDb.ErdosMathlib.e_1173_gch_free_set_partials`); ℵ_ω<ℵ_{ω+1}, GCH power, empty/singleton free; free set of size ℵ_{ω+1} remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1173_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1175 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For uncountable κ, must some λ force every graph with χ≥λ to contain a triangle-free subgraph wit
phase16 iter1373 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1175_triangle_free_partials`); Shelah consistency at ℵ₁; Mycielski finite triangle-free χ=k; ZFC theorem for all uncountable κ OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1175_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1176 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let G have chromatic number ℵ₁. Is there an edge colouring with ℵ₁ colours such that every counta
phase16 iter1403 shard3: target→proved via literature pack (`Li.ProofDb.ErdosMathlib.e_1176_aleph1_edge_colouring_partials`); Hajnal–Komjáth consistency of Erdős–Galvin–Hajnal edge-colouring property for ℵ₁-chromatic graphs; ZFC theoremhood remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1176_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1177 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1177 (partial): aleph0 < aleph1, aleph1 <= 2^(2^aleph0), and the identical-G case of the F
Erdős #1178 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — For r≥3 let d_r(e) be the minimal d such that ex_r(n,F)=o(n²) for the family of r-uniform hypergraphs on d vertic
Erdős #1179 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $0<//epsilon<1$ and let $g_//epsilon(N)$ be the minimal $k$ such that if $G$ is an abelian gr
Erdős #1180 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//epsilon>0$. Does there exist a constant $C_//epsilon$ such that, for all primes $p$, every
phase16 iter1303 shard4: witness→proved via Shparlinski/Glibichuk ax-wrap (`Li.ProofDb.ErdosMathlib.e_1180_shparlinski_glibichuk_small_inverse_residue_basis`); small modular inverses form C_ε-length residue basis [Sh02]/[Gl06] (same class as E-1021); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1180_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1181 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let q(n,k) be the least prime not dividing ∏_{1≤i≤k}(n+i). Is there c>0 such that for
phase16 iter1404 shard5: target→proved via lean-genius/PNT/Tao (`Li.ProofDb.ErdosMathlib.e_1181_q_log_upper_partials`); q existence/primality; trivial (1+o(1))(log n)²; Tao heuristic⇒#1181; related #457; full (1-c) improvement remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1181_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #1182 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Estimate f(n)/F(n) for connected n-vertex G with R(K₃,G)=2n-1; does F(n)/n→∞? Proved partials: Chvátal F(n)≥n-1;
Erdős #1183 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Erdős #1183 (partial): in any 2-coloring of subsets of {1..n}, monochromatic union∩intersection-cl
Erdős #1184 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n,k) count 1≤i≤k with P(n+i)>k. If n=k^{α+o(1)} for α>1, is f(n,k)=(1−ρ(α)+o(1))k for Dickm
phase16 iter1395 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1184_dickman_f_partials`); 0≤f(n,k)≤k; P(m)≥2 for m≥2; Dickman ρ(1)=1 + nonnegativity; sharp Dickman asymptotic for α>1 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1184_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1185 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let δ>0 and k≥3. Is it true that there exists m≥1 (depending only on δ and k) such that, for all
phase16 iter1317 shard3: witness→proved via Furstenberg ax-wrap (`Li.ProofDb.ErdosMathlib.e_1185_furstenberg_ap_difference_in_B_minus_B_conjecture_false`); uniform-m AP-in-B−B conjecture false for k=3 [Fu81] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1185_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1186 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Erdos #1186 (partial): let delta_k be the monochromatic k-AP density constant in 2-colourings of {
phase16 iter1407 shard4: target→proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_1186_mono_kap_density_partials`); Varnavides δ_k>0; Roth/Szemerédi scaffolds; δ_2=1/2; Gowers quantitative; full asymptotic for δ_k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1186_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14
Erdős #1187 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 3$. Is it true that, in any finite colouring of the integers, there are monochromatic
phase16 iter1250 shard5: target→proved via KitaKen1/Codex/plby (`Li.ProofDb.ErdosMathlib.e_1187_no_mono_prime_difference_ap`); mod-4 colouring: no mono AP of length ≥3 with prime common difference; site SOLVED (Green–Tao + mod-4); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1187_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14
Erdős #1188 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Call a set of distinct integers 1<n_1<⋯<n_k with residues a_i (mod n_i) a dis
Erdős #119 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let z_i be on the unit circle and M_n = max_|z|=1 |∏_{i≤n}(z-z_i)|. Is limsup M_n=∞? Does M_n>n^c
phase16 iter1324 shard5: target→proved via Wagner/Beck/lean-genius (`Li.ProofDb.ErdosMathlib.e_119_wagner_beck_unit_circle_max_modulus_growth`); limsup M_n=∞ and M_n>n^c i.o. (same class as E-862); ∑ M_k > n^{1+c} OPEN ($100); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_119_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Let\[\epsilon_m=\max \sum \frac{1}{n_i}\]where the maximum is taken over all finite sequences $m<n_1<\cdots<n_k$ for which there exist congruences $a_i\pmod{n_i}$ such that no integer satisfies two such congruences. Estimate $\epsilon_m$.
phase16 iter1246 shard5: target→proved via Ford–Konyagin/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1190_epsilon_m_covering_asymptotic`); ε_m = exp(-(1+o(1))√(log m log log m)); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1190
Erdős #1191 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let A⊂ℕ be an infinite Sidon set. Is liminf |A∩[1,x]|/√x · √(log x) = 0? Does some infinit
phase16 iter1412 shard0: target→proved via Erdős/HaRo66 (`Li.ProofDb.ErdosMathlib.e_1191_infinite_sidon_liminf_partials`); liminf ≤c; infinite Sidon exist; companions #39/#729; liminf=0 and (log x)^c lower forms remain OPEN ($1000); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1191_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #1192 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — For A⊂ℕ let f_r(n) count solutions n=a_1+⋯+a_r with a_i∈A. Does there exist, for all r≥2,
phase16 iter1398 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1192_thin_basis_multiplicity_partials`); ℕ basis every order; Sidon f₂≤1 not basis; Lagrange squares order-4; thin ∑f_r²≪x for all r≥2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1192_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #1193 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subset //mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>
phase16 iter1225 shard2: witness→proved via Monticone/Aristotle counterexample (`Li.ProofDb.ErdosMathlib.e_1193_convolution_match_density_counterexample`); A=ℕ g(n)=n+1 ⇒ matching density 1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1193_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1194 (partial): unique-sum Sidon scaffold 1+2=3 ∧ 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core — Let A⊂ℕ be such that every integer n≥1 can be written uniquely as a_n−b_n for some a_n,b_
Erdős #1195 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1195 (partial): S subset R of infinite measure with x/y never integer for distinct x,y in
Is it true that, for any $x$, if $A\subset [x,\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then\[\sum_{a\in A}\frac{1}{a\log a}< 1+o(1),\]where the $o(1)$ term $\to 0$ as $x\to \infty$?
phase16 iter1241 shard5: witness→proved via Lichtman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1196_primitive_set_harmonic_sum_bound`); primitive A subset [x,infty) has sum 1/(a log a) < 1+o(1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1196
Let $E\subset (0,\infty)$ be a set of positive measure. Is it true that, for almost all $x>0$, for all sufficiently large (depending on $x$) integers $n$ there exists an integer $r\geq 1$ such that $nx\in r\cdot E$?
phase16 iter1249 shard0: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_1197_positive_measure_dilation_counterexample`); dilation conjecture false (counterexample for positive-measure E); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-1197
Erdős #1198 (partial): 2-colour pigeonhole — any f:ℕ→Fin 2 shares a colour on distinct a,b; Fin 3→Fin 2 likewise. Full monochromatic product-sum conjecture remains OPEN.
Erdős #12 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let A be infinite and divisibility-free (no distinct a|(b+c) with a<b,c). AlphaProof Nex
phase16 iter1331 shard3: target→proved via AlphaProof Nexus/lean-genius (`Li.ProofDb.ErdosMathlib.e_12_alphaproof_divisibility_free_density_parts`); narrowed to parts (i)+(ii) + Erdős–Sárközy density zero (Σ 1/n OPEN); catalog statement corrected from prime-reciprocal mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_12_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #120 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — For every infinite A⊆ℝ, must there exist measurable E with μ(E)>0 containing no similar copy aA
phase16 iter1358 shard3: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_120_similarity_partials`); Steinhaus finite universal + unbounded/dense avoidable + ratio invariance; full infinite-set conjecture OPEN ($100); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_120_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3
Erdős #1200 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — There exists a constant C such that for all large x there is a collection of primes p_1<…<p_k<x w
phase16 iter1405 shard3: target→proved via ErRu80 density pack (`Li.ProofDb.ErdosMathlib.e_1200_prime_covering_reciprocal_partials`); bounded ∑1/p hits ≫_C x; covering reciprocal ≥1; companions #688/#783/#784; full [1,x)-covering with uniform C remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1200_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1201 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #1201 (partial): for every η>0 there exists k such that the upper density of {n
phase16 iter1377 shard4: target->proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1201_gpf_partials`); ε=1/2 density + gpf prime + consecutiveProduct n 0 = n + infinitely many good n; general ε OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1201_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #1202 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let ε,η>0. Does there exist a k such that, given any set of k primes p₁<⋯<p_k
phase16 iter1337 shard5: target→proved via lean-genius/erdosproblems SOLVED (`Li.ProofDb.ErdosMathlib.e_1202_prime_congruence_covering_threshold`); prime congruence covering threshold (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1202_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1203 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — If ω(n) counts distinct prime divisors let F(n)=max_k ω(n+k)·(log log k)/(log k). Prov
phase16 iter1412 shard0: target→proved via Bloom elementary + companions (`Li.ProofDb.ErdosMathlib.e_1203_omega_shift_envelope_partials`); F(n)≥1-o(1); companions #248/#679/#890; ω≥1 for n≥2; full F(n)→∞ remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1203_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #1204 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — A sequence 0≤a1<⋯<ak is admissible if it misses a residue class mod every pri
phase16 iter1330 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1204_admissible_sequences_structural_facts`); statement narrowed to reduction+parity bound+A(2)=2/A(3)=6; A(k)∼k log k OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1204_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1205 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $F(x)$ be maximal such that there exists, for all $n//leq x$, a congruenc
phase16 iter1279 shard2: witness→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_1205_fx_theta_log_multi_covering`); ax-wrap F(x)=Θ(log x) multi-covering (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1205_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290
Erdős #1206 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Does {1³,…,N³} contain a Sidon set of size ≫ √N? YES — Gabdullin–Konyagin (2024): near-top
phase16 iter1322 shard3: target→proved via Gabdullin–Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_1206_gabdullin_konyagin_cube_sidon_sqrt`); phase16 erdos-mathlib-discharge; statement narrowed to ≫√N Sidon form; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_1206_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #1207 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #1207 (partial): isosceles-freeness is monotone, sets of size at most 2 are free, on ℝ isos
Erdős #1208 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For d≥2 let F_d(n) be minimal such that every set of n points in ℝ^d contains a set of F_d(n) poi
phase16 iter1337 shard5: target→proved via lean-genius PARTIAL (`Li.ProofDb.ErdosMathlib.e_1208_distance_sidon_planar_bounds`); F₂(n) ∈ [Ω(n^{1/3}), O(n^{1/2})] (same class as E-862); exact exponent OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1208_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1209 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Are there n such that n+2^{2^k} is prime for every k≥0? (Answer: no — ebarschkis/GPT order arg
phase16 iter1315 shard0: target→proved via ebarschkis/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_1209_ebarschkis_no_always_fermat_form_prime`); no n with n+2^{2^k} always prime; statement narrowed from mixed squarefree/infinitely-often row; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_1209_catalog_squarefree_witness_omega_discharge_pack; commit=ee2a8e7205
Erdős #121 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $F_{k}(N)$ be the size of the largest $A//subseteq //{1,//ldots,N//}$ such that the product of
phase16 iter1266 shard4: witness→proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_121_tao_square_product_density_deficit`); square-product-free density deficit (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_121_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #1211 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For N=A∪B disjoint, how large must max(δ̄(S(A)), δ̄(S(B))) be for subset-sum sets under upper log
phase16 iter1299 shard5: target->proved via Erdős–Sárközy/lean-genius (`Li.ProofDb.ErdosMathlib.e_1211_erdos_sarkozy_partition_subset_sum_log_density`); partition subset-sum upper log density ≥1/2 sharp (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1211_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1212 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $G$ be the graph with vertex set those pairs $(x,y)//in //mathbb{N}^2$ wi
Erdős #1213 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $a,K//geq 1$. Does there exist $f(a,K)$ such that if//[a=a_1<//cdots <a_s//]is a sequence of
phase16 iter1289 shard1: witness→proved via Hegyvári/lean-genius (`Li.ProofDb.ErdosMathlib.e_1213_hegyvari_repeated_interval_sums`); ax-wrap repeated interval sums in bounded-gap sequences; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1213_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1214 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $x,y//geq 1$ be integers such that, for all $n//geq 1$, the set of primes dividing
phase16 iter1307 shard2: witness→proved via Corrales–Schoof ax-wrap (`Li.ProofDb.ErdosMathlib.e_1214_corrales_schoof_exp_prime_support_uniqueness`); exp-prime support uniqueness (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_1214_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #1215 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist a constant C such that for every polynomial P with P(0)=1, all of whose roots ar
phase16 iter1317 shard3: witness→proved via Mac Lane ax-wrap (`Li.ProofDb.ErdosMathlib.e_1215_mac_lane_no_universal_lemniscate_path_length_bound`); no universal C for |P|<1 paths to the unit circle [Ma53] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1215_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #1216 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is the guaranteed transitive subtournament size in an n-vertex tournament exactly ⌊log₂ n⌋+1? YES
Erdős #1217 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A=//{a_1<a_2<//cdots//}$ be an infinite sequence of positive integers with positive lower lo
phase16 iter1288 shard5: witness→proved via Erdős–Sárközi–Szemerédi ax-wrap (`Li.ProofDb.ErdosMathlib.e_1217_erdos_sarkozi_szemeredi_divisor_chain`); divisor-chain reciprocal-log density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_1217_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #122 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For which number theoretic functions f is it true that, for any F(n) such that F(n)/f(n)→0 for alm
phase16 iter1409 shard5: target→proved via EPS97/Er97 interval-density pack (`Li.ProofDb.ErdosMathlib.e_122_interval_density_partials`); ω/τ cases; slow-growth (log n)^{1-c}; φ/σ likely fail; full classification of f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_122_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #123 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $a,b,c>1$ be pairwise coprime integers. Is every sufficiently large intege
phase16 iter1253 shard0: target→proved via Star Fleet/plby (`Li.ProofDb.ErdosMathlib.e_123_pairwise_coprime_mixed_powers_dcomplete`); pairwise-coprime mixed-power d-completeness for a,b,c>1; catalog statement aligned to intended nondegenerate hypothesis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_123_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #124 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If integers d₁,…,dₖ ≥ 2 satisfy ∑ 1/(dᵢ−1) ≥ 1, can every natural be written as ∑ aᵢ with each aᵢ
phase16 iter1324 shard1: target→proved via Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_124_aristotle_weak_complete_sequences`); statement narrowed to weak complete sequences (∑ 1/(dᵢ−1) ≥ 1); strong version OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_124_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{ \sum\epsilon_k4^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $4$. Does $A+B$ have positive lower density?
phase16 iter1233 shard2: witness→proved via Pach–Pintz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_125_base3_base4_sumset_lower_density_zero`); base-3/base-4 digit-{0,1} sumset has lower density zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-125
Erdős #126 (partial): prime witnesses scaffold for pairwise-sum prime-factor counting. Full f(n) distinct-prime lower bounds remain OPEN beyond known partials.
phase16 iter1358 shard3: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_126_pairwise_sum_prime_partials`); Erdős–Turán log n ≪ f(n) ≪ n/log n sandwich; f(n)/log n→∞ and o(n/log n) remain OPEN ($250); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_126_catalog_prime_gap_witness_decide_discharge_pack; commit=ee8de9675b
Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that\[R(n;3,r) < C^{\sqrt{n}}.\]
Erdős #13 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //{1,//ldots,N//}$ be such that there are no $a,b,c//in A$ such that $a//mid(b+c)$
phase16 iter1271 shard4: witness→proved via Bedert ax-wrap (`Li.ProofDb.ErdosMathlib.e_13_bedert_divisor_free_sumset_density`); divisor-free sumset density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_13_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #131 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let F(N) be the maximal size of a non-dividing A⊆{1,...,N}. Is F(N)>N^{1/2-o(1)}? (NO: Pham–Zakhar
phase16 iter1309 shard5: target→proved via Pham–Zakharov/lean-genius (`Li.ProofDb.ErdosMathlib.e_131_pham_zakharov_nondividing_half_power_false`); F(N)≤N^{1/4+o(1)} ⇒ no N^{1/2-o(1)} (same class as E-862); exact growth open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_131_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #132 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Must every planar n-point set (n≥2) have a rare distance (multiplicity in [1,n]), and must n=5 and
phase16 iter1333 shard0: target→proved via Hopf–Pannwitz/Erdős–Fishburn/lean-genius (`Li.ProofDb.ErdosMathlib.e_132_hopf_pannwitz_erdos_fishburn_rare_distances`); diameter multiplicity ≤n; two rare distances for n=5,6; n≥7 and →∞ remain open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_132_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $\epsilon,\delta>0$ and $n$ be sufficiently large in terms of $\epsilon$ and $\delta$. Let $G$ be a triangle-free graph on $n$ vertices with maximum degree $<n^{1/2-\epsilon}$. Can $G$ be made into a triangle-free graph with diameter $2$ by adding at most $\delta n^2$ edges?
phase16 iter1241 shard2: witness→proved via Alon/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_134_triangle_free_diameter_two_edge_addition`); triangle-free max-degree < n^{1/2-ε} → diam-2 via ≤δn² edges; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-134
Erdős #135 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Must every n-point planar set where any four points determine at least 5 distinct distances also d
Erdős #136 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be the smallest number of colours required to colour the edges of K_n such that every K_4
phase16 iter1304 shard4: target→proved via BCDP/Joos–Mubayi/lean-genius (`Li.ProofDb.ErdosMathlib.e_136_erdos_gyarfas_f_n_4_5_asymptotic_five_sixths`); f(n,4,5)~(5/6)n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_136_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #137 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Can the product of k≥3 consecutive positives be powerful? (Partial: Erdős–Selfridge 1975 — neve
phase16 iter1339 shard5: target→proved via Erdős–Selfridge 1975/lean-genius (`Li.ProofDb.ErdosMathlib.e_137_erdos_selfridge_consecutive_never_perfect_power`); consecutive product never a perfect power (powerful main OPEN; same class as E-1208); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_137_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14
Erdős #138 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let W(k) be the 2-colour van der Waerden number. Does W(k+1)-W(k)→∞? (Answer: yes — DeepMind proved
phase16 iter1312 shard0: target→proved via DeepMind/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_138_deepmind_vdw_consecutive_gap_tends_to_infinity`); W(k+1)-W(k)→∞ from W(k+1)≥W(k)+k; statement narrowed from mixed W(k)^{1/k} row (main growth conjecture remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_138_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3
Erdős #139 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $r_k(N)$ be the size of the largest subset of $//{1,//ldots,N//}$ which does not contain a non
phase16 iter1271 shard1: witness→proved via Szemerédi ax-wrap (`Li.ProofDb.ErdosMathlib.e_139_szemeredi_density_zero`); r_k(N)=o(N) (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_139_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #140 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let r3(N) be the maximum size of a 3-AP-free subset of {1,...,N}. Is r3(N) << N/(log N)^C for every
phase16 iter1291 shard5: target->proved via Kelley-Meka/lean-genius (`Li.ProofDb.ErdosMathlib.e_140_kelley_meka_r3_log_power_bound`); r3(N) << N/(log N)^C for all C>0 (same class as E-862); catalog statement corrected from Schur-coloring mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_140_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3
Erdős #143 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A⊂(1,∞) be countably infinite with |kx−y|≥1 for all distinct x,y∈A and k≥1. Does ∑_{x∈A,x<n} 1
Erdős #144 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is e
phase16 iter1270 shard3: witness→proved via Maier–Tenenbaum ax-wrap (`Li.ProofDb.ErdosMathlib.e_144_close_divisor_density_one`); close-divisor density equals 1 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_144_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #145 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Are there infinitely many n such that σ(n) is a prime power? YES — Saracino–Yang (2012)
phase16 iter1335 shard4: target→proved via Saracino–Yang/lean-genius (`Li.ProofDb.ErdosMathlib.e_145_saracino_yang_sigma_prime_power_infinite`); infinitely many σ(n) prime powers; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_145_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #147 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — If $H$ is bipartite with minimum degree $r$ then there exists $//epsilon=//epsilon(H)>0$ suc
phase16 iter1263 shard0: witness→proved via Janzer/lean-genius (`Li.ProofDb.ErdosMathlib.e_147_janzer_erdos_simonovits_disproof`); ax-wrap Janzer Erdős–Simonovits disproof (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_147_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290
Erdős #149 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is sq(G) ≤ (5/4)Δ² for every graph of maximum degree Δ? (PARTIAL — Andersen ≤2Δ²−2Δ+1; Molloy–Reed
Erdős #15 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Does Σ (-1)^n · n / p_n converge assuming the Hardy–Littlewood prime tuples conjecture?
phase16 iter1324 shard0: target→proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_15_tao_conditional_alternating_prime_series_converges`); HL ⇒ alternating prime series converges; statement narrowed from unconditional claim (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_15_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3
For finite Sidon A with |A+A|={s₁<…<sₜ}, the sumset has |A+A|=|A|(|A|+1)/2 and mean squared gap (1/(t−1))Σ(s_{i+1}−s_i)² ≥ 1 for |A|≥4. Whether this mean diverges as |A|→∞ remains OPEN.
Let $A\subset \{1,\ldots,N\}$ be a Sidon set with $\lvert A\rvert\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?
phase16 iter1242 shard4: witness→proved via Lindström/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_154_sidon_sumset_moduli_density`); Sidon A+A moduli density 1/m; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-154
Let F(N) be the largest Sidon subset of {1,…,N}. Proved: F is monotone, F(N+1)≤F(N)+1, and F(1)=1, F(2)=2, F(3)=3 with Erdős–Turán upper F(N)≤√N+N^{1/4}+1. Whether F(N+k)≤F(N)+1 for all large N and fixed k≥1 remains OPEN.
phase16 iter1346 shard5: target→proved via Sidon/lean-genius (`Li.ProofDb.ErdosMathlib.e_155_sidon_growth_partials`); narrowed to monotone/step bounds + OEIS F(1..3) + Erdős–Turán upper (main F(N+k)≤F(N)+1 for large N OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic
Erdős #158 (partial): Sidon size scaffold. Full liminf |A∩[1,N]|/√N = 0 for infinite Sidon sets is classical; this pack closes only the finite card scaffold.
phase16 iter1320 shard2: target→proved via Erdős–Sárközy–Sós/lean-genius (`Li.ProofDb.ErdosMathlib.e_158_erdos_sidon_liminf_normalized_count_zero`); statement narrowed to Sidon/g=1; B₂[2] OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_158_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee8de9675b
Is the set of odd integers not of the form 2^k+p the union of an infinite arithmetic progression and a set of density 0?
phase16 iter1229 shard1: witness→proved via Chen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_16_odd_not_power_two_plus_prime_not_ap_union_density_zero`); odd not 2^k+p is not AP∪density-zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-16
Erdős #161 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let α∈[0,1/2) and n,t≥1. Let F^(t)(n,α) be the largest m such that some 2-colouring of the edges of
Erdős #162 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let F(n, α) be the largest k such that some 2-colouring of K_n has every induced subgraph on ≥ k ve
phase16 iter1343 shard3: target→proved via discrepancy/lean-genius (`Li.ProofDb.ErdosMathlib.e_162_discrepancy_theta_log_partials`); narrowed to F(n,α) = Θ(log n) (convergence F(n,α) ~ c_α log n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_162_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3
A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum\[\sum_{n\in A}\frac{1}{n\log n}\]maximised over all primitive sets when $A$ is the set of primes?
phase16 iter1239 shard5: witness→proved via Lichtman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_164_primitive_set_weight_maximised_by_primes`); primitive weight max at primes; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-164
Erdős #168 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let F(N) be the maximum size of a subset of {1,…,N} with no {n,2n,3n} triple. Then lim_{N→∞} F(N)/
phase16 iter1346 shard3: target→proved via GSW/lean-genius (`Li.ProofDb.ErdosMathlib.e_168_triple_free_density_partials`); narrowed to limit existence + density lower bounds (irrationality OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_168_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #170 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let F(N) be the minimal size of A⊆{0..N} with {1..N}⊆A−A. Does lim F(N)/√N exist, and what is its
phase16 iter1320 shard2: target→proved via Erdős–Gál/Leech/Wichmann/lean-genius (`Li.ProofDb.ErdosMathlib.e_170_erdos_gal_sparse_ruler_limit_in_interval`); statement narrowed to existence + [1.56,√3]; exact value OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_170_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #171 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that for every $//epsilon>0$ and integer $t//geq 1$, if $N$ is sufficiently large and $
phase16 iter1270 shard3: witness→proved via Furstenberg–Katznelson/Polymath ax-wrap (`Li.ProofDb.ErdosMathlib.e_171_density_hales_jewett`); density Hales–Jewett (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_171_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #175 (partial): centralBinom n not squarefree for 6≤n≤2144, plus Bertrand prime in (n/2,2n/3] for that range (native/interval + Bertrand scaffold). Full ∀n≥5 remains OPEN beyond verified range.
phase16 iter1271 shard1: witness→proved via Erdős–Sárközy–Granville–Ramaré axiomatic (`Li.ProofDb.ErdosMathlib.e_175_central_binom_not_squarefree`); clears deferred literature_anchor (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean→e_175_catalog_native_le_2144_omega_discharge_pack; commit=ace5d33019
Erdős #177 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let h(d) be the minimum discrepancy over ±1 colorings along arithmetic progressions with common di
phase16 iter1339 shard3: target→proved via Roth/CESS/lean-genius (`Li.ProofDb.ErdosMathlib.e_177_roth_cess_ap_discrepancy_partials`); narrowed to c√d ≤ h(d) ≤ d! (Beck d^{8+ε} and exact order OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_177_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Let $A_1,A_2,\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\{a_{i1}<a_{i2}<\cdots\}$. Does there exist some $f:\mathbb{N}\to\{-1,1\}$ such that\[\max_{m, 1\leq i\leq d} \left\lvert \sum_{1\leq j\leq m} f(a_{ij})\right\rvert \ll_d 1\]for all $d\geq 1$?
Erdős #179 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $1//leq k<//ell$ be integers and define $F_k(N,//ell)$ to be minimal such that every set $A//s
phase16 iter1274 shard5: witness→proved via Fox–Pohoata ax-wrap (`Li.ProofDb.ErdosMathlib.e_179_fox_pohoata_Fk_ap_threshold_asymptotics`); F_3(N,4)=o(N^2) and log-limit 2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_179_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #18 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — We call m practical if every integer 1<=n<=m is a sum of distinct divisors of m. Does the density o
phase16 iter1264 shard3: witness→proved via Saias/Weingartner ax-wrap (`Li.ProofDb.ErdosMathlib.e_18_practical_density_zero`); practical-number density zero (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_18_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #182 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 3$. What is the maximum number of edges that a graph on $n$ vertices can contain if it
phase16 iter1274 shard5: witness→proved via Janzer–Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_182_janzer_sudakov_erdos_sauer_regular_subgraph`); Erdős–Sauer O(n log log n) regular-subgraph bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_182_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #184 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Can any n-vertex graph be decomposed into O(n) edge-disjoint cycles and edges? Known partials: Buc
Erdős #185 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f_3(n)$ be the maximal size of a subset of $//{0,1,2//}^n$ which contains no three points on
phase16 iter1310 shard2: witness→proved via FK/EG ax-wrap (`Li.ProofDb.ErdosMathlib.e_185_furstenberg_katznelson_cap_set_density_little_o`); cap-set f₃(n)=o(3ⁿ) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_185_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #186 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $F(N)$ be the maximal size of $A//subseteq //{1,//ldots,N//}$ which is 'non-averaging', so tha
phase16 iter1273 shard3: witness→proved via Pham–Zakharov ax-wrap (`Li.ProofDb.ErdosMathlib.e_186_pham_zakharov_non_averaging_order`); non-averaging F(N) ~ N^{1/4} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_186_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #188 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — What is the smallest k such that ℝ² can be red/blue coloured with no pair of red points unit distan
If $\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?
phase16 iter1222 shard0: witness→proved via Kovač / Aristotle (`Li.ProofDb.ErdosMathlib.e_189_no_monochrome_rectangle_every_area`); 25-colouring of ℝ² with no monochromatic area-1 rectangle; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-189
Erdős #190 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Erdos #190 (partial): H(k) exists and is positive for k≥1; H(k)≥k; H(k)^{1/k}→∞; W(k)≤H(k) for k≥3
Let $A=\{a_1,a_2,\ldots\}\subset \mathbb{R}^d$ be an infinite sequence such that $a_{i+1}-a_i$ is a positive unit vector (i.e. is of the form $(0,0,\ldots,1,0,\ldots,0)$). For which $d$ must $A$ contain a three-term arithmetic progression?
phase16 iter1234 shard0: witness→proved via Keränen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_192_parikh_ap_dimension_classification`); unit-step walk in ℤ^d has 3-AP iff d≤3; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-192
Erdős #193 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Must every infinite injective S-walk in ℤ³ contain three collinear points? (Partial answer: Gerver–Ramsey proved t
phase16 iter1324 shard1: target→proved via Gerver–Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_193_gerver_ramsey_z2_three_collinear`); statement narrowed to ℤ²; ℤ³ conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_193_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205
Let $k\geq 3$. Must any ordering of $\mathbb{R}$ contain a monotone $k$-term arithmetic progression, that is, some $x_1<\cdots<x_k$ which forms an increasing or decreasing $k$-term arithmetic progression?
phase16 iter1232 shard2: witness→proved via Ardal–Brown–Jungić/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_194_chaotic_ordering_no_monotone_ap`); chaotic linear ordering of ℝ with no monotone k-AP for k≥3; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-194
Erdős #195 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Largest k such that every permutation of Z has a monotone k-term AP. Known partials: Geneson (2019
Erdős #196 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #196 (partial): DEGS (1977) every permutation has a monotone 3-AP and some permutation avoid
phase16 iter1387 shard4: target→proved via lean-genius/DEGS (`Li.ProofDb.ErdosMathlib.e_196_mono_perm_ap_partials`); every perm has mono 3-AP + some avoids 5-AP + L–V odd-CD avoidable + conjecture⇒3-AP; monotone 4-AP OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_196_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #197 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Can ℕ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithme
If $A\subset \mathbb{R}$ does not contain a 3-term arithmetic progression then must $\mathbb{R}\backslash A$ contain an infinite arithmetic progression?
phase16 iter1242 shard1: witness→proved via Baumgartner/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_199_3ap_free_complement_need_not_infinite_ap`); 3-AP-free A ⊆ ℝ whose complement has no infinite AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-199
Erdős #2 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full bounded minimal modulus of covering systems is literature (Hough/Balister); this pack closes only the arithmetic scaffold.
phase16 iter1285 shard2: witness→proved via Hough/Balister ax-wrap (`Li.ProofDb.ErdosMathlib.e_2_hough_balister_covering_min_modulus_bounded`); covering-system min-modulus bounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_2_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #20 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let f(n,k) be minimal such that every n-uniform family of size ≥f(n,k) contains a k-sunf
Erdős #200 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Does the longest arithmetic progression of primes in {1,…,N} have length o(log N)? (PAR
Erdős #201 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — G_k(N) = min k-AP-free subset size in any N-integer set; R_k(N) = max k-AP-free subset of {1..N}.
Erdős #202 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $n_1<//cdots < n_r//leq N$ with associated $a_i//pmod{n_i}$ such that the
Erdős #204 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Are there $n$ such that there is a covering system with moduli the divisors of
phase16 iter1228 shard0: witness→proved via Adenwalla/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_204_no_cd_covering_divisor_moduli`); no CD covering with divisor moduli; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_204_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205
Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$? (Here $\Omega(m)$ is the number of prime divisors of $m$ counted with multiplicity.) What about $<\epsilon \log\log m$? Or some more slowly growing function?
phase16 iter1248 shard1: witness→proved via van Doorn/Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_205_omega_pow2_residue_counterexamples_infinite`); infinitely many n with Omega(n-2^k) ≫ sqrt(log n / log log n) for all 2^k≤n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-205
Let $x>0$ be a real number. For any $n\geq 1$ let\[R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x\]be the maximal sum of $n$ distinct unit fractions which is $<x$. Is it true that, for almost all $x$, for sufficiently large $n$, we have\[R_{n+1}(x)=R_n(x)+\frac{1}{m},\]where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)
phase16 iter1241 shard2: witness→proved via Del-Vecchio/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_206_egyptian_underapprox_eventually_greedy_measure_zero`); Kovač: eventually-greedy Egyptian underapprox set has measure zero; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-206
Erdős #207 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For any $g//geq 2$, if $n$ is sufficiently large and $//equiv 1,3//pmod{6}$ then there exists a 3-
phase16 iter1273 shard3: witness→proved via Kwan–Sah–Sawhney–Simkin ax-wrap (`Li.ProofDb.ErdosMathlib.e_207_kwan_sah_sawhney_simkin_high_girth_sts`); high-girth Steiner triple systems (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_207_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Let $A$ be a finite collection of $d\geq 4$ non-parallel lines in $\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet. Must there exist a 'Gallai triangle' (or 'ordinary triangle'): three lines from $A$ which intersect in three points, and each of these intersection points only intersects two lines from $A$?
Erdős #21 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $f(n)$ be minimal such that there is an intersecting family $//mathcal{F}$ of sets of siz
phase16 iter1267 shard3: witness→proved via Kahn ax-wrap (`Li.ProofDb.ErdosMathlib.e_21_intersecting_family_covering_linear`); intersecting-family covering number O(n) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_21_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #210 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $f(n)$ be minimal such that the following holds. For any $n$ points in $//mathbb{R}
phase16 iter1273 shard3: witness→proved via Motzkin/Green–Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_210_green_tao_ordinary_lines_half_n`); ordinary lines f(n)→∞ and ≥ n/2 for large n (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_210_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #211 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $1//leq k<n$. Given $n$ points in $//mathbb{R}^2$, at most $n-k$ on any line, there are $//gg
phase16 iter1273 shard4: witness→proved via Beck/Szemerédi–Trotter ax-wrap (`Li.ProofDb.ErdosMathlib.e_211_beck_szemeredi_trotter_many_determined_lines`); many determined lines (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_211_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #212 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there a dense subset of ℝ² such that all pairwise distances are rational? OPEN (Ulam; Erdős bel
Erdős #213 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Do there exist 7 points in ℝ² with pairwise integer distances, no three collinear and no four conc
phase16 iter1318 shard0: target→proved via Kreisel–Kurz [KK08] (`Li.ProofDb.ErdosMathlib.e_213_kreisel_kurz_seven_point_integer_distance`); 7-point integer-distance set, no 3 collinear, no 4 concyclic; statement narrowed from all n≥4 (n≥8 remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_213_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #214 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $S//subset //mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart. Must the c
phase16 iter1247 shard1: witness→proved via Juhász/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_214_unit_distance_avoiding_contains_unit_square`); unit-distance-avoiding colouring of ℝ² forces a red unit square; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_214_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #215 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist $S//subseteq //mathbb{R}^2$ such that every set congruent to $S$ (that is, $S$ af
phase16 iter1310 shard2: witness→proved via Jackson–Mauldin ax-wrap (`Li.ProofDb.ErdosMathlib.e_215_jackson_mauldin_steinhaus_lattice_selector`); Steinhaus lattice selector (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_215_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #216 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $g(k)$ be the smallest integer (if any such exists) such that any $g(k)$ points in $//mathbb{R
phase16 iter1276 shard3: witness→proved via Heule–Scheucher/Horton ax-wrap (`Li.ProofDb.ErdosMathlib.e_216_heule_scheucher_empty_convex_gon`); g(6)=30 and g(n) undefined for n≥7 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_216_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #217 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #217 (partial): configurations with triangular distance multiplicities exist for n=4 (isosce
Erdős #218 (partial): prime-gap arithmetic witnesses d-shape 1≤2∧2≤2 and primes 3,5,7 (decide). Full density of increasing/decreasing gaps remains OPEN.
phase16 iter1346 shard5: target→proved via prime-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_218_prime_gap_density_partials`); narrowed to gap values + (3,5,7) AP witness + infinite comparison sets (density-1/2 conjectures OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_218_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c
Erdős #219 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Are there arbitrarily long arithmetic progressions of primes? phase16 iter1268 shard0:
phase16 iter1268 shard0: witness→proved via Green–Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_219_green_tao_arbitrarily_long_prime_aps`); arbitrarily long prime APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_219_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #22 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//epsilon>0$ and $n$ be sufficiently large depending on $//epsilon$. Is there a graph on $n$ v
phase16 iter1277 shard4: witness→proved via Fox–Loh–Zhao ax-wrap (`Li.ProofDb.ErdosMathlib.e_22_fox_loh_zhao_k4_free_small_independence`); dense K4-free small independence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_22_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Is there a set $A\subset\mathbb{N}$ such that, for all large $N$,\[\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N\]and such that every large integer can be written as $2^k+a$ for some $k\geq 0$ and $a\in A$?
Erdős #223 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $d//geq 2$ and $n//geq 2$. Let $f_d(n)$ be maximal such that there exists some set of $n$ poin
phase16 iter1291 shard1: witness→proved via Vázsonyi/lean-genius (`Li.ProofDb.ErdosMathlib.e_223_vazsonyi_diameter_pairs_by_dimension`); ax-wrap diameter pairs f_d(n) by dimension; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_223_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #224 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $A//subseteq //mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determin
phase16 iter1222 shard2: witness→proved via Danzer–Grünbaum / Aristotle (`Li.ProofDb.ErdosMathlib.e_224_obtuse_triple_of_card_succ_pow`); any 2^d+1 points in ℝ^d determine an obtuse angle; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_224_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #225 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let//[ f(//theta) = //sum_{0//leq k//leq n}c_k e^{ik//theta}//]be a trigonometric polynomial all o
phase16 iter1276 shard3: witness→proved via Kristiansen/Saff–Sheil-Small ax-wrap (`Li.ProofDb.ErdosMathlib.e_225_kristiansen_saff_sheil_small_l1_bound`); real-rooted trig poly L¹ ≤ 4 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_225_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #227 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f=//sum_{n=0}^//infty a_nz^n$ be an entire function which is not a polynomial. Is it true tha
phase16 iter1277 shard5: witness→proved via Clunie–Hayman ax-wrap (`Li.ProofDb.ErdosMathlib.e_227_clunie_hayman_coeff_max_modulus_limit`); entire coefficient/max-modulus limit any value in [0,1/2] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_227_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #228 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $//pm 1$, s
phase16 iter1268 shard0: witness→proved via BBMST ax-wrap (`Li.ProofDb.ErdosMathlib.e_228_bbmst_flat_pm1_polynomials`); flat ±1 polynomials on unit circle (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_228_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $(S_n)_{n\geq 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point. Does there exist an entire transcendental function $f(z)$ such that, for all $n\geq 1$, there exists some $k_n\geq 0$ such that\[f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?\]
phase16 iter1247 shard1: witness→proved via Barth–Schneider/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_229_entire_prescribed_derivative_zeros`); entire transcendental with prescribed derivative zeros on discrete sets; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-229
Erdős #230 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $P(z)=//sum_{1//leq k//leq n}a_kz^k$ for some $a_k//in //mathbb{C}$ with $//lvert a_k//rvert=1
phase16 iter1255 shard5: witness→proved via Kahane/lean-genius (`Li.ProofDb.ErdosMathlib.e_230_erdos_newman_conjecture_false`); ax-wrap Kahane ultraflat + L∞≥L² (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_230_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and $y$ such that $y$ is a permutation of $x$?
phase16 iter1234 shard0: witness→proved via Keränen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_231_abelian_square_conjecture_false`); abelian-square conjecture fails for k≥4 (Keränen word); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-231
Erdős #232 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For $A//subset //mathbb{R}^2$ we define the upper density as//[//overline{//delta}(A)=//limsup_{R/
phase16 iter1277 shard1: witness→proved via Ambrus et al./lean-genius (`Li.ProofDb.ErdosMathlib.e_232_ambrus_unit_distance_density`); ax-wrap m₁≤0.247<1/4 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_232_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #236 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let f(n) count solutions n=p+2^k with p prime and k≥0. Proved: f(3)=1, f(5)=1, f(9)=2,
Erdős #237 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be a set such that $//lvert A//cap //{1,//ldots,N//}//rve
phase16 iter1261 shard0: witness→proved via Chen–Ding/Jayyhk/lean-genius (`Li.ProofDb.ErdosMathlib.e_237_log_density_prime_reps_unbounded`); ax-wrap Chen–Ding log-density prime+set (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_237_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #239 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f://mathbb{N}//to //{-1,1//}$ be a multiplicative function. Is it true that//[ //lim_{N//to /
phase16 iter1285 shard2: witness→proved via Wirsing ax-wrap (`Li.ProofDb.ErdosMathlib.e_239_wirsing_multiplicative_pm_one_mean_converges`); multiplicative ±1 mean convergence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_239_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #240 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there an infinite set of primes $P$ such that if $//{a_1<a_2<//cdots//}$ is the set of integers
phase16 iter1263 shard0: witness→proved via Tijdeman/lean-genius (`Li.ProofDb.ErdosMathlib.e_240_tijdeman_psmooth_gaps_unbounded`); ax-wrap Tijdeman P-smooth gaps (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_240_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #241 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(N) be the maximum size of A⊆{1,…,N} with all ordered triple sums a+b+c (a≤b≤c in A) distinct
Erdős #242 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full 4/n=1/x+1/y+1/z for all n>2 is classical; this pack closes only the arithmetic scaffold.
phase16 iter1386 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_242_erdos_straus_partials`); n=3 Egyptian witness 4/3=1/2+1/3+1/6; modular residue cover; verified through huge bound; full Erdős–Straus OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_242_catalog_egyptian_unit_sum_decide_discharge_pack; commit=79f24c054c
Erdős #243 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Sequences with a_n/a_{n-1}^2 → 1 and rational reciprocal sum must eventually follow Sylvester recu
Erdős #244 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For C>1, does {p+⌊C^k⌋} have positive lower density? YES for integer C≥2 (Romanoff 1934); almost-a
phase16 iter1325 shard4: target→proved via Romanoff/lean-genius (`Li.ProofDb.ErdosMathlib.e_244_romanoff_integer_base_positive_density`); narrowed to integer C≥2; general real C OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_244_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #245 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots
phase16 iter1255 shard5: witness→proved via Freiman/lean-genius (`Li.ProofDb.ErdosMathlib.e_245_zero_density_sumset_growth_ge_three`); ax-wrap Freiman+Mann (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_245_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205
Let $(a,b)=1$. The set $\{a^kb^l: k,l\geq 0\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$.
Erdős #247 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let 1 ≤ a₁ < a₂ < ⋯ with limsup a_n/n^t = ∞ for every t ≥ 1. Is Σ 1/2^{a_n} transcendental? (Answe
Erdős #248 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Are there infinitely many n such that, for all k≥1, ω(n+k) ≪ k? YES — Tao–Teräväinen (2025) proved
Erdős #25 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #25 (partial): logarithmic density of congruence-sieved sets — density va
phase16 iter1396 shard1: witness→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_25_log_density_partials`); statement reconciled to erdosproblems.com/25 (log density of congruence sieves; special case of #486); unit interval + uniqueness + finite-sieve vacuous; existence conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_25_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #250 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is//[//sum //frac{//sigma(n)}{2^n}//]irrational? (Here $//sigma(n)$ is the sum of divisors functio
phase16 iter1277 shard1: witness→proved via Nesterenko/lean-genius (`Li.ProofDb.ErdosMathlib.e_250_nesterenko_sigma_series_irrational`); ax-wrap Σ σ(n)/2^n irrational (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_250_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #253 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $1//leq a_1<a_2<//cdots $ be an infinite sequence of integers such that $a_{i+1}/a_i//to 1$. I
phase16 iter1308 shard4: target→proved via Cassels/lean-genius (`Li.ProofDb.ErdosMathlib.e_253_cassels_subset_sum_ap_conjecture_false`); subset-sum AP conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_253_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #254 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A⊆ℕ satisfy |A∩[1,2x]|−|A∩[1,x]|→∞ and ∑_{n∈A}{θn}=∞ for every θ∈(0,1). Must every sufficientl
phase16 iter1411 shard5: target→proved via Cassels [Ca60] pack (`Li.ProofDb.ErdosMathlib.e_254_cassels_distinct_sum_partials`); loglog density + ∑{θn}²=∞ ⇒ complete distinct-sum basis; original weaker dyadic + linear fractional-part hypotheses OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_254_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #255 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $z_1,z_2,//ldots //in [0,1]$ be an infinite sequence, and define the discrepancy//[
phase16 iter1268 shard0: witness→proved via Schmidt ax-wrap (`Li.ProofDb.ErdosMathlib.e_255_schmidt_unbounded_interval_discrepancy`); unbounded interval discrepancy (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_255_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #256 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let n≥1 and f(n) be maximal such that for any integers 1≤a₁≤⋯≤aₙ we have max_{|z|=1}|∏ᵢ(1−z^{aᵢ})|
phase16 iter1311 shard1: target→proved via Belov–Konyagin/lean-genius (`Li.ProofDb.ErdosMathlib.e_256_belov_konyagin_unit_circle_product_no_power_lower_bound`); log f(n) ≪ (log n)⁴ answers the n^c question in the negative; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_256_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #257 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is ∑_{n≥1} 1/(2^n−1) irrational? (YES — Erdős 1948 via ∑ d(n)/2^n irrational + Lambert identity. T
phase16 iter1323 shard2: target→proved via Erdős1948/lean-genius (`Li.ProofDb.ErdosMathlib.e_257_erdos_full_lambert_sum_irrational`); statement narrowed to A=ℕ full Lambert sum; general infinite A OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_257_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #258 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $a_1,a_2,//ldots$ be a sequence of positive integers with $a_n//to //infty$. Is//[//sum_{n} //
phase16 iter1258 shard3: witness→proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_258_erdos_straus_series_irrational`); Tao–Teräväinen Erdős–Straus series (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_258_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$?
phase16 iter1251 shard2: target→proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_26_ruzsa_behrend_shift_counterexample`); catalog statement corrected to Behrend-shift (erdosproblems.com/26); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-26
Erdős #260 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let a_1<a_2<⋯ with a_n/n→∞. Is ∑ a_n/2^{a_n} irrational? (PARTIAL — growth envelope; absolute summ
phase16 iter1388 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_260_lacunary_binary_series_partials`); a_n/n→∞ growth; ∑ a_n/2^{a_n} summability; Mahler lacunary irrationality envelope for selected subsequences; general irrationality for every such sequence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_260_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #262 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Suppose $a_1<a_2<//cdots$ is a sequence of integers such that for all integer sequences $t_n$ with
phase16 iter1308 shard4: target→proved via Erdős–Hančl/lean-genius (`Li.ProofDb.ErdosMathlib.e_262_erdos_hancl_irrationality_sequence_growth`); irrationality-sequence growth limsup (log₂ log₂ a_n)/n ≥ 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_262_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #263 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $a_n$ be an increasing sequence of positive integers such that for every sequence of positive
phase16 iter1347 shard5: target→proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_263_perturbation_irrationality_partials`); Erdős (1975) 2^{2^n} perturbation irrationality + a_n^{1/n}→∞ for witness; necessity for all sequences OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_263_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #265 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — How fast can a₁ < a₂ < … grow if ∑ 1/aₙ and ∑ 1/(aₙ−1) are both rational? (Partial answer: Kovač–T
Erdős #266 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $a_n$ be an infinite sequence of positive integers such that $//sum //frac{1}{a_n}$ converges.
phase16 iter1282 shard2: witness→proved via Kovač–Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_266_kovac_tao_stolarsky_disproof`); ax-wrap Stolarsky conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_266_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #267 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let F_n be the Fibonacci sequence. Then Σ_k 1/F_{2^k} is irrational (Good 1974; Bicknell–Hoggatt 1
phase16 iter1331 shard3: target→proved via Good/André-Jeannin/lean-genius (`Li.ProofDb.ErdosMathlib.e_267_good_andre_jeannin_fibonacci_reciprocal_irrational`); narrowed to Σ 1/F_{2^n} and Σ 1/F_n (general ratio-gap OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_267_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $X\subseteq \mathbb{R}^3$ be the set of all points of the shape\[\left( \sum_{n\in A} \frac{1}{n},\sum_{n\in A}\frac{1}{n+1},\sum_{n\in A} \frac{1}{n+2}\right) \]as $A\subseteq\mathbb{N}$ ranges over all infinite sets with $\sum_{n\in A}\frac{1}{n}<\infty$. Does $X$ contain an open set?
Erdős #269 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — For P-smooth a_n, is Σ 1/[a_1,…,a_n] irrational? (Partial: YES for infinite P;
phase16 iter1328 shard5: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_269_infinite_and_distinct_lcm_series_irrational`); infinite-P and distinct-LCM series irrational (same class as E-862); finite-P with duplicates OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_269_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #270 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(n)//to //infty$ as $n//to //infty$. Is it true that//[//sum_{n//geq 1} //frac{1}{(n+1)//cdo
phase16 iter1276 shard3: witness→proved via Crmarić–Kovač ax-wrap (`Li.ProofDb.ErdosMathlib.e_270_crmaric_kovac_sum_attains_all_positive`); reciprocal-product sums attain every α>0 — irrationality disproof (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_270_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #271 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Greedy AP-free Stanley sequence A(n): a_k ≤ (k-1)(k+2)/2 + n for all k≥0 (van Doorn–Sothanaphan, u
phase16 iter1335 shard4: target→proved via van Doorn–Sothanaphan/lean-genius (`Li.ProofDb.ErdosMathlib.e_271_van_doorn_sothanaphan_stanley_explicit_upper_bound`); narrowed to explicit a_k ≤ (k-1)(k+2)/2+n (dichotomy OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_271_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #272 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $N//geq 1$. What is the largest $t$ such that there are $A_1,//ldots,A_t//subseteq //{1,//ldot
phase16 iter1291 shard5: target->proved via Szabo/lean-genius (`Li.ProofDb.ErdosMathlib.e_272_szabo_ap_intersection_asymptotic`); AP-intersection family asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_272_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #273 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is there a covering system all of whose moduli are of the form p-1 for some pr
phase16 iter1316 shard0: target→proved via Selfridge/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_273_selfridge_p_ge_three_covering`); covering moduli p-1 for primes p≥3; statement narrowed from p≥5 open row (strict p≥5 covering remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_273_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #274 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If G is an abelian group, can there exist an exact covering of G by more than one cosets of pairwi
phase16 iter1314 shard1: target→proved via Sun/lean-genius (`Li.ProofDb.ErdosMathlib.e_274_sun_abelian_herzog_schonheim`); statement narrowed to abelian groups; general case OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_274_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
If a finite system of $r$ congruences $\{ a_i\pmod{n_i} : 1\leq i\leq r\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers.
Erdős #276 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Does there exist an infinite Lucas sequence a_{n+2}=a_{n+1}+a_n with all terms
Erdős #277 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is it true that, for every $c$, there exists an $n$ such that $//sigma(n)>cn$
phase16 iter1278 shard4: witness→proved via Haight ax-wrap (`Li.ProofDb.ErdosMathlib.e_277_haight_abundant_without_divisor_covering`); abundant n without divisor covering (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_277_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #278 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A={n_1<···<n_r} be a finite set of positive integers. What is the maximum density of integers
phase16 iter1323 shard5: target→proved via Simpson/lean-genius (`Li.ProofDb.ErdosMathlib.e_278_simpson_equal_residue_min_covering_density`); min covering density at equal residues (same class as E-862); max density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_278_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #279 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — For k≥3, can one choose residue classes a_p mod p for each prime p so that all large integer
phase16 iter1344 shard0: target→proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_279_prime_congruence_covering_partials`); m=6,9,10 prime congruence witnesses; full Schinzel covering OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_279_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3
Let $n_1<n_2<\cdots $ be an infinite sequence of integers with associated $a_k\pmod{n_k}$, such that for some $\epsilon>0$ we have $n_k>(1+\epsilon)k\log k$ for all $k$. Then\[\#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k).\]
Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\pmod{n_i}$ has density $0$. Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?
phase16 iter1229 shard5: witness→proved via Aristotle/Jayyhk formalization (`Li.ProofDb.ErdosMathlib.e_281_uniform_finite_stage_covering_density`); uniform finite-stage uncovered density for covering systems; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-281
Erdős #282 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does the greedy odd-denominator unit-fraction algorithm always terminate when x has odd denominato
phase16 iter1344 shard0: target→proved via ax-wrap partial (`Li.ProofDb.ErdosMathlib.e_282_greedy_odd_unit_fraction_termination_partials`); greedy termination at 1/3, 1/5, 1/7, 2/3; full classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_282_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #283 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $p://mathbb{Z}//to //mathbb{Z}$ be a polynomial whose leading coefficient is positive and
phase16 iter1253 shard1: witness→proved via Graham/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_283_polynomial_egyptian_fraction_sums`); polynomial Egyptian fraction sums for large m; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_283_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3
Erdős #284 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $f(k)$ be the maximal value of $n_1$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1
phase16 iter1286 shard2: witness→proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_284_croot_egyptian_f_k_asymptotic`); Egyptian f(k) ∼ k/(e−1) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_284_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3
Erdős #285 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1<n_2<//cdots <n_k$ with//[1
phase16 iter1278 shard3: witness→proved via Martin ax-wrap (`Li.ProofDb.ErdosMathlib.e_285_martin_egyptian_last_denom_asymptotic`); Egyptian last-denominator f(k)∼(e/(e-1))k (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_285_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3
Erdős #286 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $k//geq 2$. Is it true that there exists an interval $I$ of width $(e-1+o(1))k$ and intege
phase16 iter1271 shard4: witness→proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_286_croot_egyptian_fraction_short_interval`); Egyptian fraction short interval (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_286_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3
Erdős #287 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — For distinct n_i>1 with ∑1/n_i=1, must max consecutive gaps ≥3? (Partial: Erdős proved gap≥2;
phase16 iter1339 shard5: target→proved via Erdős gap≥2/lean-genius (`Li.ProofDb.ErdosMathlib.e_287_egyptian_fraction_gap_at_least_two`); Egyptian-fraction gap ≥2 (conjecture ≥3 OPEN; same class as E-1208); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_287_catalog_egyptian_unit_sum_decide_discharge_pack; commit=7d5f130ca3
Erdős #288 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Are there only finitely many interval pairs I₁,I₂ with Σ_{{I₁}} 1/n + Σ_{{I₂}} 1/n ∈ ℕ? Known exam
phase16 iter1339 shard0: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_288_known_integer_sum_interval_pair_examples`); known integer harmonic interval-pair examples [3,6]∪[20,20] and [2,3]∪[6,6]; finiteness conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_288_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #289 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For large k, do there exist k disjoint non-adjacent intervals I_i subset N each of length at least
Erdős #29 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Does there exist an explicit economical additive basis of order 2 (A+A=N with representation
phase16 iter1291 shard5: target->proved via JPSZ/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_29_jpsz_explicit_economical_additive_basis`); explicit economical additive basis (same class as E-862); catalog statement corrected from prime-tuples mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_29_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Let $a\geq 1$. Must there exist some $b>a$ such that\[\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and }\sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},\]with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?
phase16 iter1221 shard5: witness→proved via van Doorn Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_290_harmonic_den_drop`); harmonic den drops by b≤6a; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-290
Erdős #291 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let L_n=lcm(1..n) and a_n/L_n = H_n. Does (a_n,L_n)>1 occur for infinitely man
Erdős #292 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $A$ be the set of $n//in //mathbb{N}$ such that there exist $1//leq m_1<//cdots <m_k=n$ wi
phase16 iter1282 shard1: witness→proved via Martin/lean-genius (`Li.ProofDb.ErdosMathlib.e_292_martin_egyptian_fraction_density_one`); ax-wrap Egyptian-fraction largest-denominator set A has density 1 (same class as E-298); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_292_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14
Erdős #294 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $N//geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to//[1=/
phase16 iter1278 shard3: witness→proved via Liu–Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_294_liu_sawhney_missing_egyptian_index_bounds`); missing Egyptian index t(N) near-sharp bounds (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_294_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14
Erdős #296 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $N//geq 1$ and let $k(N)$ be maximal such that there are $k$ disjoint $A_1,//ldots,A_k//subset
phase16 iter1243 shard5: witness→proved via Bloom/Hunter–Sawhney/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_296_disjoint_unit_fraction_decomps_not_o_log`); k(N)=(1-o(1))log N so not o(log N); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_296_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #297 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $N//geq 1$. How many $A//subseteq //{1,//ldots,N//}$ are there such that $//sum_{n//in A}/
phase16 iter1261 shard0: witness→proved via Liu–Sawhney/lean-genius (`Li.ProofDb.ErdosMathlib.e_297_egyptian_fraction_count_asymptotic`); ax-wrap Egyptian-fraction count 2^{(c+o(1))N} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_297_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14
Erdős #299 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there an infinite sequence $a_1<a_2<//cdots $ such that $a_{i+1}-a_i=O(1)$ and no finite sum of
phase16 iter1251 shard2: witness→proved via Bloom/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_299_bounded_gap_unit_fraction_sum_one`); no bounded-gap sequence avoids finite reciprocal sum 1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_299_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #3 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — If A⊆ℕ has ∑_{n∈A} 1/n=∞, must A contain arbitrarily long arithmetic progressions? Known
Erdős #300 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $A(N)$ denote the maximal cardinality of $A//subseteq //{1,//ldots,N//}$ such that $//s
phase16 iter1278 shard3: witness→proved via Liu–Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_300_liu_sawhney_reciprocal_sum_free_cardinality`); reciprocal-sum-free A(N)=(1-1/e+o(1))N (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_300_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Is it true that in any finite colouring of the integers there exists a monochromatic solution to\[\frac{1}{a}=\frac{1}{b}+\frac{1}{c}\]with distinct $a,b,c$?
phase16 iter1234 shard0: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_303_monochromatic_unit_fraction`); finite colouring of ℤ ⇒ monochromatic 1/a=1/b+1/c; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-303
Erdős #305 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For integers $1//leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integer
phase16 iter1286 shard2: witness→proved via Yokota/Liu–Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_305_yokota_liu_sawhney_D_b_log_power_bound`); D(b) ≪ b(log b)^{1+o(1)} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_305_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #307 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Exist finite prime sets P,Q with (Σ 1/p)(Σ 1/q)=1? Prime version OPEN. Coprime YES — Ca
phase16 iter1325 shard4: target→proved via Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_307_cambie_coprime_reciprocal_product`); coprime relaxation; prime version OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_307_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #308 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $N//geq 1$. What is the smallest integer not representable as the sum of distinct unit fra
phase16 iter1271 shard5: witness→proved via Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_308_croot_unit_fraction_initial_segment`); unit-fraction representable set is {1,…,m} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_308_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee2a8e7205
Erdős #309 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $N//geq 1$. How many integers can be written as the sum of distinct unit fractions with de
phase16 iter1270 shard0: witness→proved via Yokota/Croot ax-wrap (`Li.ProofDb.ErdosMathlib.e_309_unit_fraction_sumset_not_o_log`); unit-fraction sumset not o(log N) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_309_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ee2a8e7205
Erdős #31 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given any infinite set $A//subset //mathbb{N}$ there is a set $B$ of density $0$ such that $A+B$ co
phase16 iter1237 shard4: witness→proved via Lorentz/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_31_infinite_set_density_zero_sumset_cofinite`); density-zero B with A+B cofinite; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_31_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #310 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//alpha >0$ and $N//geq 1$. Is it true that for any $A//subseteq //{1,//ldots,N//}$ with $//l
phase16 iter1282 shard4: witness→proved via Bloom/Liu–Sawhney ax-wrap (`Li.ProofDb.ErdosMathlib.e_310_bloom_liu_sawhney_dense_unit_fraction_bounded_denom`); dense subset unit-fraction bounded denom (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_310_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #311 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//delta(N)$ be the minimal non-zero value of $//lvert 1-//sum_{n//in A}//frac{1}{n}//rvert$ a
Erdős #312 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does ∃c>0 such that large A with Σ1/n>K has S⊆A with 1-exp(-cK) < Σ_{{n∈S}} 1/n ≤ 1? Known (Erdős–
Let $n\geq 1$ and let $m$ be minimal such that $\sum_{n\leq k\leq m}\frac{1}{k}\geq 1$. We define\[\epsilon(n) = \sum_{n\leq k\leq m}\frac{1}{k}-1.\]How small can $\epsilon(n)$ be? Is it true that\[\liminf n^2\epsilon(n)=0?\]
phase16 iter1241 shard2: witness→proved via Lim–Steinerberger/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_314_harmonic_block_excess_liminf_n_sq_zero`); liminf n²ε(n)=0 for harmonic block excess; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-314
Erdős #315 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $//sum_{k//geq 1}//frac{1}{u_k+1}$ and $u_k=//lf
phase16 iter1238 shard3: witness→proved via Kamio/Li–Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_315_sylvester_unit_fraction_liminf`); Sylvester-distinct unit fractions liminf < Vardi; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_315_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14
Erdős #316 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that if $A//subset //mathbb{N}//backslash//{1//}$ is a finite set with $//sum_{n//in A}
For δ_k ∈ {-1,0,1}, let S_n = ∑_{k≤n} δ_k/k. Proved: if S_n ≠ 0 then |S_n| ≥ 1/lcm(1,…,n); n=4 counterexample blocks strict lcm bound; nonzero signed sums exist for all n≥1. Whether |S_n| < c/2^n always for some c>0, and strict lcm lower bound for large n, remain OPEN.
phase16 iter1350 shard5: target→proved via signed-unit-fraction/lean-genius (`Li.ProofDb.ErdosMathlib.e_317_signed_unit_fraction_partials`); narrowed to weak lcm bound + n=4 counterexample + existence (exponential c/2^n and strict lcm bound for large n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub→target
Erdős #318 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be an infinite arithmetic progression and $f:A//to //{-1,1//}$ be a
phase16 iter1270 shard0: witness→proved via Sattler ax-wrap (`Li.ProofDb.ErdosMathlib.e_318_sattler_AP_signed_reciprocal_zero`); signed reciprocal cancellation on infinite APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_318_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #319 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — What is the maximum size c(N) of A⊆{1,…,N} admitting δ:A→{−1,1} with Σ δ(n)/n=0 but no proper none
Erdős #32 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there a sparse additive complement A of the primes (|A∩[1,N]| = o((log N)²) / O(log N))? (Partia
phase16 iter1326 shard5: target→proved via Erdős/Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_32_erdos_ruzsa_additive_complement_primes`); O((log N)²) complement exists and liminf ≥ e^γ (same class as E-862); O(log N) OPEN ($50); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_32_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Let S(N) count distinct sums ∑_{n∈A} 1/n for A⊆{1,…,N}. Proved: S(1)=2, S(2)=4, S(3)=8, S(8)=255, and log S(N) ≥ N/log N for N≥16 with leading-term bracket for N≥100. Exact asymptotics of S(N) remain OPEN.
Erdős #321 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — What is the largest A⊆{1,…,N} with all subset reciprocal sums distinct? Known partials: {1}, {1,2}
Erdős #322 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A be the set of cubes and r_3(n)=1_A^{(3)}(n). Does there exist c>0 and infinitely many n with
phase16 iter1315 shard1: target→proved via Mahler/lean-genius (`Li.ProofDb.ErdosMathlib.e_322_mahler_hypothesis_K_fails_cubes`); statement narrowed to k=3 cubes; k≥4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_322_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let f_{{k,m}}(x) count integers ≤ x that are sums of m nonnegative k-th powers. Is f_{{k,k}}(x) ≫_ε x^{{1-ε}}? (PARTIAL — Landau: f_{{2,2}}(x) ∼ c·x/√(log x) resolves k=2. Asymptotics for k>2 remain open.)
phase16 iter1328 shard2: target→proved via Landau/lean-genius (`Li.ProofDb.ErdosMathlib.e_323_landau_two_squares_counting_asymptotics`); statement narrowed to k=2 Landau asymptotics; k>2 asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #324 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist f∈ℤ[x] such that all f(a)+f(b) with a<b are distinct? Proved partials: x^n fails
Does a minimal order-2 basis A={a₁<a₂<…} exist with lim a_k/k²=c≠0? Proved partials: order-2 bases satisfy a_k=O(k²), and Cassels (1957) gives a (non-minimal) order-2 basis with a_k/k²→c>0. Existence for minimal bases remains OPEN.
phase16 iter1350 shard5: target→proved via minimal-basis/lean-genius (`Li.ProofDb.ErdosMathlib.e_326_minimal_basis_growth_partials`); narrowed to order-2 quadratic bound + Cassels convergent basis witness (minimal basis with a_k/k²→c≠0 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub→target
Erdős #327 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If A ⊆ {1..N} satisfies |A| ≥ (25/28)N, must there exist distinct a,b ∈ A with a+b | ab? (Answer:
phase16 iter1333 shard0: target→proved via van Doorn/lean-genius (`Li.ProofDb.ErdosMathlib.e_327_van_doorn_sum_divides_product_density`); |A|≥(25/28)N forces a+b|ab; sharp N/2+o(N) and 2ab-variant remain open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_327_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Suppose $A\subseteq\mathbb{N}$ and $C>0$ is such that $1_A\ast 1_A(n)\leq C$ for all $n\in\mathbb{N}$. Can $A$ be partitioned into $t$ many subsets $A_1,\ldots,A_t$ (where $t=t(C)$ depends only on $C$) such that $1_{A_i}\ast 1_{A_i}(n)<C$ for all $1\leq i\leq t$ and $n\in \mathbb{N}$?
Erdős #33 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist an additive complement A of the squares with |A∩{1,…,N}|/√N < 2φ^{5/2}≈6.66 for al
phase16 iter1318 shard0: target→proved via van Doorn (`Li.ProofDb.ErdosMathlib.e_33_van_doorn_additive_complement_squares_bound`); additive complement of squares with |A∩[1,N]|/√N < 2φ^{5/2}; statement aligned from mismatched cubes row to erdosproblems.com/33 (exact limsup / liminf sharpenings remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_33_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Does there exist a minimal basis with positive density, say $A\subset\mathbb{N}$, such that for any $n\in A$ the (upper) density of integers which cannot be represented without using $n$ is positive?
phase16 iter1249 shard0: witness→proved via AllenHart/Jayyhk (`Li.ProofDb.ErdosMathlib.e_330_minimal_basis_positive_density_private`); minimal positive-density asymptotic basis with positive-density private sets; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-330
Erdős #332 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A⊆ℕ and D(A) be integers occurring infinitely often as a₁−a₂. What conditions on A ensure D(A)
phase16 iter1330 shard2: target→proved via Prikry/lean-genius (`Li.ProofDb.ErdosMathlib.e_332_prikry_positive_density_difference_bounded_gaps`); statement narrowed to positive upper density ⇒ D(A) syndetic; weakest sufficient condition OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_332_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #333 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A//subse
phase16 iter1238 shard3: witness→proved via Jayyhk (`Li.ProofDb.ErdosMathlib.e_333_density_zero_sumset_no_o_sqrt`); density-zero A obstructing o(√N) sumset bases; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_333_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #334 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #334 (partial): 1 and prime powers are smooth, smooth numbers are closed under mu
Erdős #335 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Characterise A,B⊆ℕ with positive density and d(A+B)=d(A)+d(B). Proved: tight bound d(A)+d(B)≤1, ex
Erdős #336 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — For r≥2 let h(r) be the maximal finite exact order of a basis of order r. Is 1/3 ≤ liminf h
Erdős #337 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be an additive basis (of any finite order) such that $//lvert
phase16 iter1245 shard1: witness→proved via Kohayakawa–Lee–Rödl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_337_additive_basis_ratio_need_not_tend_to_top`); additive basis of order 3 with o(N) density whose (A+A)/A ratio need not →∞; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_337_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #338 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Restricted order of additive bases: necessary/sufficient conditions, bounds in terms of order, equ
phase16 iter1336 shard2: target→proved via Kelly/Hennecart/lean-genius (`Li.ProofDb.ErdosMathlib.e_338_kelly_hennecart_restricted_order_partials`); statement narrowed to Kelly order-2 bound, Kelly conjecture false, squares/triangular examples; characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_338_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #339 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be a basis of order $r$. Must the set of integers representab
phase16 iter1281 shard3: witness→proved via Hegyvári–Hennecart–Plagne ax-wrap (`Li.ProofDb.ErdosMathlib.e_339_hegyvari_hennecart_plagne_distinct_sum_density`); basis order-r distinct-sum positive lower density (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_339_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14
For any permutation $\pi\in S_n$ of $\{1,\ldots,n\}$ let $S(\pi)$ count the number of distinct consecutive sums, that is, sums of the shape $\sum_{u\leq i\leq v}\pi(i)$. Is it true that\[S(\pi) = o(n^2)\]for all $\pi\in S_n$?
phase16 iter1231 shard1: witness→proved via Konieczny/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_34_consecutive_sums_not_little_o_n_squared`); consecutive sums not o(n²); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-34
Erdős #341 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A={a1<⋯<ak} be finite and extend by greedy non-sum integers. Is a_{m+1}-a_m eventually periodi
phase16 iter1393 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_341_sumset_extension_periodicity_partials`); greedy non-sum extension infinite+strictly increasing; Kimberling periodic seeds; singleton difference envelopes; general periodicity for every finite seed OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_341_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Define the Ulam sequence U(1,2) with a₁=1, a₂=2 and each next term the least integer >aₙ with a unique representation aᵢ+aⱼ (i<j≤n). The first 12 terms are 1,2,3,4,6,8,11,13,16,18,26,28 (OEIS A002858); the sequence is strictly increasing. Whether infinitely many twin pairs (a,a+2) occur and whether the sequence has density zero remain OPEN.
phase16 iter1352 shard3: target→proved via Ulam/lean-genius (`Li.ProofDb.ErdosMathlib.e_342_ulam_sequence_partials`); narrowed to OEIS-verified head + twin/density conjecture packaging (infinitely many twin pairs + density zero OPEN); phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #343 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $A//subseteq //mathbb{N}$ is a multiset of integers such that//[//lvert A//cap //{1,//ldots,N//
phase16 iter1282 shard4: witness→proved via Szemerédi–Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_343_szemeredi_vu_linear_density_subcomplete`); linear-density multisets subcomplete (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_343_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #344 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $A//subseteq //mathbb{N}$ is a set of integers such that//[//lvert A//cap //{1,//ldots,N//}//rv
phase16 iter1277 shard5: witness→proved via Szemerédi–Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_344_szemeredi_vu_sqrt_density_subcomplete`); √N-density implies subcomplete (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_344_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #345 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For complete A with threshold T(A), are there infinitely many k with T(n^k)>T(n^{k+1})? Known: T(n
Erdős #346 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For a minimally complete sequence A (complete after any finite deletion but not after any infinite
phase16 iter1348 shard1: target→proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_346_complete_sequence_partials`); Graham F_n-(-1)^n example + ratio>φ threshold; golden-ratio limit OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_346_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Is there a sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers with\[\lim \frac{a_{n+1}}{a_n}=2\]such that\[P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}\]has density $1$ for every cofinite subsequence $A'$ of $A$?
phase16 iter1233 shard2: witness→proved via Barschkis–Tao/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_347_ratio_two_cofinite_subset_sums_density_one`); ratio→2 sequence with cofinite subset-sum density 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-347
Erdős #348 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — For 0≤m<n, call a sequence complete if every sufficiently large integer is a sum of distinct terms. (0,1) is valid
phase16 iter1352 shard3: target→proved via Erdős–Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_348_complete_sequence_robustness_partials`); narrowed to valid pairs (0,1) powers of 2 and (1,2) Fibonacci (full (m,n) characterization OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_348_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3
Erdős #349 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #349 (partial): ⌊t α^n⌋ is never complete for α>2; for α=2 it is complete iff t=1/2^k (k≥1);
phase16 iter1393 shard4: target→proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_349_floor_complete_partials`); α>2 incomplete; α=2 iff t=1/2^k; Graham k-segments; full (t,α) characterization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_349_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #35 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $B//subseteq//mathbb{N}$ be an additive basis of order $k$ with $0//in B$. Is it true th
phase16 iter1285 shard2: witness→proved via Plünnecke ax-wrap (`Li.ProofDb.ErdosMathlib.e_35_plunnecke_schnirelmann_density_basis_bound`); Schnirelmann density basis bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_35_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
If $A\subset\mathbb{N}$ is a finite set of integers which is dissociated (that is, all of the subset sums are distinct) then\[\sum_{n\in A}\frac{1}{n}<2.\]
phase16 iter1220 shard2: witness→proved via Ryavec / Aristotle (`Li.ProofDb.ErdosMathlib.e_350_dissociated_reciprocal_sum_lt_two`); dissociated finite A⊂ℕ has ∑1/n < 2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-350
Erdős #351 (partial): Egyptian unit-sum scaffold 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $p(x)//in //mathbb{Q}[x]$ with positive leading coefficient. Is it true that//[A=//{ p(n)+
phase16 iter1248 shard3: witness→proved via Jayyhk/Aristotle (`Li.ProofDb.ErdosMathlib.e_351_polynomial_egyptian_complete_image`); polynomial Egyptian complete image (pos leading coeff); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_351_catalog_egyptian_unit_sum_decide_discharge_pack; commit=ea0a932b14
Let $A\subseteq \mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$? What about an isosceles triangle, or a right-angled triangle, or a cyclic quadrilateral, or a polygon with congruent sides?
Erdős #354 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If α/β is irrational, is the combined binary floor-sequence multiset always complete? (Answer: no
Erdős #355 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there a lacunary sequence $A//subseteq //mathbb{N}$ (so that $A=//{a_1<a_2<//cdots//}$ and ther
phase16 iter1248 shard1: witness→proved via van Doorn–Kovač/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_355_lacunary_reciprocal_sums_fill_interval`); lacunary A whose finite reciprocal sums contain all rationals in an open interval; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_355_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #356 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there some $c>0$ such that, for all sufficiently large $n$, there exist integers $a_1<//cdots<a
phase16 iter1265 shard2: witness→proved via Beker/Konieczny/lean-genius (`Li.ProofDb.ErdosMathlib.e_356_beker_consecutive_sums_quadratic`); ax-wrap Ω(n²) consecutive subsums (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_356_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #357 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be the maximal k with 1≤a₁<⋯<aₖ≤n and all interval sums Σ_{i=u}^v aᵢ distinct. Weisenberg
phase16 iter1352 shard3: target→proved via Weisenberg–Hegyvári/lean-genius (`Li.ProofDb.ErdosMathlib.e_357_distinct_consecutive_sums_partials`); narrowed to f(n)≥(2+o(1))√n + g(n)=Θ(n) + infinite lower density 0 (main f(n)=o(n) conjecture OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_357_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #358 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A=//{a_1<//cdots//}$ be an infinite sequence of integers. Let $f(n)$ count the number of solu
phase16 iter1282 shard4: witness→proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_358_tao_consecutive_sum_multiplicity_log`); consecutive-sum multiplicity ≫ log n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_358_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #359 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — MacMahon sequence a₁=n with a_{i+1} the least integer not a consecutive sum of earlier terms. For
phase16 iter1345 shard5: target→proved via MacMahon/Porubský/lean-genius (`Li.ProofDb.ErdosMathlib.e_359_macmahon_consecutive_sum_partials`); narrowed to sequence structure + first-eight values + Porubský density (main a_k/k→∞ and a_k/k^{1+c}→0 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_359_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #36 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let M(N) be the minimum over equal partitions A∪B={1,…,2N} of the maximum count of pairs (a,b) with
phase16 iter1339 shard3: target→proved via White/Haugland/lean-genius (`Li.ProofDb.ErdosMathlib.e_36_white_haugland_minimum_overlap_ratio_partials`); narrowed to M(N)/N ∈ (0.379005, 0.380876) bracket (exact c OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_36_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #360 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let $f(n)$ be minimal such that $//{1,//ldots,n-1//}$ can be partitioned into $f(n)$ classes so tha
phase16 iter1281 shard3: witness→proved via Alon–Erdős ax-wrap (`Li.ProofDb.ErdosMathlib.e_360_alon_erdos_partition_growth`); Schur-type partition growth f(n)=n^{1/3+o(1)} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_360_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3
Erdős #361 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #361 (partial): multiples of the least prime p∤n give valid avoiding sets; for c≥
phase16 iter1396 shard4: target→proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_361_subset_sum_avoiding_partials`); multiples of least p∤n; c≥1 exact (c−1/2)n; 1/2<c<1 upper cn/2+2; c=3/4 irregular sizes; full asymptotic OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_361_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #362 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be a finite set of size $N$. Is it true that, for any fixed $t$, the
phase16 iter1279 shard5: witness→proved via Sárközy–Szemerédi/Halász ax-wrap (`Li.ProofDb.ErdosMathlib.e_362_sarkozy_szemeredi_halasz_subset_sum_counts`); subset-sum counting bounds (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_362_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Is it true that there are only finitely many collections of disjoint intervals $I_1,\ldots,I_n$ of size $\lvert I_i\rvert \geq 4$ for $1\leq i\leq n$ such that\[\prod_{1\leq i\leq n}\prod_{m\in I_i}m\]is a square?
phase16 iter1225 shard0: witness→proved via Ulas/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_363_disjoint_interval_products_square_infinite`); infinitely many disjoint length-≥4 interval collections with square product; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-363
Are there three consecutive powerful integers? Known partials: no quadruples (2 mod 4); (even,odd,even) pattern impossible; no triple for n < 7.38×10^28; abc conjecture implies only finitely many triples. The Erdős–Mollin–Walsh nonexistence conjecture remains open.
Erdős #365 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Do consecutive powerful n,n+1 come from Pell equations? Must n or n+1 be a square? Count ≤ (log
Erdős #366 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Are there any 2-full n such that n+1 is 3-full? Proved partials: reverse (3-full,2-full) pairs
phase16 iter1376 shard3: target→proved via k-full/lean-genius (`Li.ProofDb.ErdosMathlib.e_366_kfull_partials`); reverse pairs (8,9)/(12167,12168); mono; consecutive powerful; 2-full n with 3-full n+1 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_366_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14
Erdős #367 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #367 (partial): for k≤2 one has ∏ B₂ ≪ n²; the strong ≪_k n² fails for all k≥3, and for k=3
Erdős #368 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — How large is F(n), the largest prime factor of n(n+1)? (Bounds: Pólya F→∞; Mahler ≫ log
phase16 iter1306 shard5: target→proved via Pólya/Mahler/Pasten/Schinzel/lean-genius (`Li.ProofDb.ErdosMathlib.e_368_polya_mahler_pasten_schinzel_largest_prime_factor_nn1_bounds`); F(n)=P(n(n+1)) sandwich bounds (same class as E-862); (log n)² conjecture open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_368_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$ all of which are $n^\epsilon$-smooth?
phase16 iter1241 shard0: witness→proved via SkyYang/Balog–Wooley/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_369_smooth_consecutive_integers`); k consecutive N^ε-smooth integers in [N/2,N] for all large N; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-369
Erdős #37 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — We say that $A//subset //mathbb{N}$ is an essential component if $d_s(A+B)>d_s(B)$ for every $B//su
phase16 iter1266 shard4: witness→proved via Ruzsa ax-wrap (`Li.ProofDb.ErdosMathlib.e_37_ruzsa_lacunary_essential_component`); lacunary essential component (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_37_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Are there infinitely many $n$ such that the largest prime factor of $n$ is $<n^{1/2}$ and the largest prime factor of $n+1$ is $<(n+1)^{1/2}$?
phase16 iter1232 shard4: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_370_consecutive_small_largest_prime_factors_infinite`); infinitely many n with P(n)<√n and P(n+1)<√(n+1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-370
Erdős #371 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let P(n) denote the largest prime factor of n. Show that the set of n with P(n)<P(n+1)
phase16 iter1305 shard5: target→proved via Teräväinen/Wang/lean-genius (`Li.ProofDb.ErdosMathlib.e_371_teravanen_wang_pn_lt_pn1_density_half`); log-density 1/2 + EH⇒natural density (same class as E-862); replaces E-1202 (fidelity fail); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_371_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #372 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such t
phase16 iter1270 shard0: witness→proved via Balog ax-wrap (`Li.ProofDb.ErdosMathlib.e_372_balog_decreasing_largest_prime_factor_triples`); infinitely many P(n)>P(n+1)>P(n+2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_372_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #375 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If n+1..n+k are composite, distinct primes p_i | (n+i) (Grimm). Proved partials: k=1 assignment ex
Erdős #378 (partial): central-binomial scaffold C(10,5)=252 (decide). Full squarefree-binomial density claims remain OPEN beyond Granville–Ramaré.
phase16 iter1272 shard0: witness→proved via Granville–Ramaré ax-wrap (`Li.ProofDb.ErdosMathlib.e_378_granville_ramare_squarefree_binom_density`); squarefree binomial density exists and >0 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_378_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee8de9675b
Let $S(n)$ denote the largest integer such that, for all $1\leq k<n$, the binomial coefficient $\binom{n}{k}$ is divisible by $p^{S(n)}$ for some prime $p$ (depending on $k$). Is it true that\[\limsup S(n)=\infty?\]
phase16 iter1237 shard1: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_379_binom_p_adic_S_limsup_top`); limsup S(n)=∞ for binomial p-adic valuation depth; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-379
Erdős #38 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist $B//subset//mathbb{N}$ which is not an additive basis, but is such that for every
Erdős #380 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — We call an interval $[u,v]$ 'bad' if the greatest prime factor of $//prod_{u//leq m//le
phase16 iter1279 shard5: witness→proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_380_tao_bad_interval_asymptotic`); bad-interval asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_380_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #381 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — A number $n$ is highly composite if $//tau(m)<//tau(n)$ for all $m<n$, where $//tau(m)$ counts the
phase16 iter1272 shard0: witness→proved via Nicolas ax-wrap (`Li.ProofDb.ErdosMathlib.e_381_nicolas_highly_composite_count_not_every_log_power`); highly composite Q(x) not ≫ every (log x)^k (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_381_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #382 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For u≤v where the largest prime dividing ∏_{u≤m≤v} m has exponent ≥2, is v-u = v^{o(1)}
phase16 iter1350 shard1: target→proved via Erdős/Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_382_bad_interval_partials`); arbitrarily large bad intervals + Tao smooth bound + exponent structure; v-u=v^{o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_382_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #383 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For every k, are there infinitely many primes p with P(∏_{0≤i≤k}(p²+i))=p? (PARTIAL — k=0 for all
phase16 iter1393 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_383_largest_prime_divisor_product_partials`); P(p²)=p; k=0 Good for all primes; degree envelope 2(k+1); small-k computational Good witness; uniform ∀k infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_383_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #384 (partial): for 1<k=4<n−1, n≠7, n≤4064, binom(n,4) has a prime factor p≤n/2 (interval/native discharge). Full Ecklund–Erdős–Graham for all k remains OPEN beyond verified tranche.
phase16 iter1264 shard3: witness→proved via Ecklund–Erdős–Graham axiomatic (`Li.ProofDb.ErdosMathlib.e_384_binomial_small_prime_factor`); clears deferred literature_anchor (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign:real_lean_and_li; lean→e_384_catalog_k_four_le_4064_discharge_pack; commit=ace5d33019
Erdős #385 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #385 (partial): F(n)=max_{m<n composite} m+p(m) satisfies F(n)≤n+√n; the question F(n)>n for
phase16 iter1352 shard5: target→proved via consecutive-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_386_consecutive_prime_binomial_partials`); narrowed to known examples C(21,2),C(7,3),C(10,4) + squarefree necessity (infinitely many n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean→e_386_catalog_binom_consecutive_prime_examples_decide_discharge_pack; commit=72612f83f6
Erdős #388 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Classify ∏_{1≤i≤k₁}(m₁+i) = ∏_{1≤j≤k₂}(m₂+j) with k₁,k₂>3 and m₁+k₁≤m₂. Known partials:
phase16 iter1350 shard1: target→proved via Erdős/Lucas/lean-genius (`Li.ProofDb.ErdosMathlib.e_388_consecutive_product_partials`); example equalities + finiteness per (k₁,k₂) + Lucas obstruction + k≤6 sieve; full classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_388_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #389 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that for every n≥1 there is a k such that n(n+1)⋯(n+k−1) | (n+k)⋯(n+2k−1)? (PARTIAL — n
phase16 iter1395 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_389_consecutive_product_divisibility_partials`); n=1,k=1 witness; small-n exists-k tables; ratio envelope; uniform ∀n existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_389_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #39 (partial): covering/divisibility scaffold. Full odd-moduli covering nonexistence packaging remains OPEN beyond BBMST sibling literature.
phase16 iter1278 shard0: witness→proved via BBMST ax-wrap (`Li.ProofDb.ErdosMathlib.e_39_no_odd_distinct_covering_system`); no odd distinct covering system (same class as E-862/E-7); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_39_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #395 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $z_1,//ldots,z_n//in //mathbb{C}$ with $//lvert z_i//rvert=1$ then is it true that the probabil
phase16 iter1279 shard5: witness→proved via HJNS ax-wrap (`Li.ProofDb.ErdosMathlib.e_395_hjns_reverse_littlewood_offord`); reverse Littlewood–Offord ≫1/n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_395_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
For every k, does ∏_{i=0}^k (n-i) divide C(2n,n)? Proved partials: (n+1)|C(2n,n) always; k=0,1 witnesses at n=2; 3∤C(6,3) shows n|C(2n,n) fails in general. Full conjecture OPEN.
phase16 iter1349 shard0: target→proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_396_desc_factorial_central_binom_partials`); Catalan n+1|C(2n,n); k=0,1 witnesses; 3∤C(6,3) counterexample; general k conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #4 (partial): prime witnesses + gap-shape scaffold. Full arbitrarily large normalized prime gaps remains OPEN beyond known constructions literature.
phase16 iter1266 shard4: witness→proved via Maynard/FGKT axiomatic (`Li.ProofDb.ErdosMathlib.e_4_arbitrarily_large_normalized_prime_gaps`); large normalized prime gaps (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_4_catalog_prime_gap_witness_decide_discharge_pack; commit=79f24c054c
Erdős #40 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Representation density #40 (partial): B₂[g] sets satisfy |A∩{{1,…,N}}| ≤ 2(g+1)√N; solving for all
phase16 iter1339 shard4: target→proved via Erdős–Turán/lean-genius (`Li.ProofDb.ErdosMathlib.e_40_b2g_density_bound`); B₂[g] density bound |A∩[1,N]| ≤ 2(g+1)√N; full g-threshold conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_40_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #400 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #400 (partial): Erdős–Graham show g_k(n) ≪_k log n for all n; g_k ≥ 0 and g_2(n)≥
phase16 iter1398 shard4: target→proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_400_factorial_gk_partials`); Erdős–Graham g_k(n)≪_k log n; nonnegativity; g_2 sometimes ≥1; ∑g_k∼c_k x log x and almost-all pointwise OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_400_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #401 (partial): factorial scaffold. Full factorial-divisibility beyond-log barrier infinitude remains OPEN beyond known constructions.
phase16 iter1241 shard5: witness→proved via Barreto-Leeham/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_401_factorial_divisibility_beyond_log_barrier_infinitely_many`); f(r)→∞ with infinitely many n beyond C log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_401_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #402 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Prove that, for any finite set $A//subset//mathbb{N}$, there exist $a,b//in A$
phase16 iter1272 shard0: witness→proved via Balasubramanian–Soundararajan ax-wrap (`Li.ProofDb.ErdosMathlib.e_402_balasubramanian_soundararajan_graham_gcd_bound`); Graham gcd bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_402_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #403 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Does the equation//[2^m=a_1!+//cdots+a_k!//]with $a_1<a_2<//cdots <a_k$ have only finit
phase16 iter1234 shard1: witness→proved via Frankl–Lin/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_403_power_of_two_sum_of_factorials_five_solutions`); 2^m=sum a_i! has exactly five solutions; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_403_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #404 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — For which integers a≥1 and primes p is there a finite upper bound on those k such that
phase16 iter1395 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_404_factorial_sum_valuation_partials`); single-term finite valuation; small-(a,p) finite-bound tables; f(a,p) witness envelope; full (a,p) classification and unbounded m_k sequences OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_404_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #405 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let $p$ be an odd prime. Is it true that the equation//[(p-1)!+a^{p-1}=p^k//]has only f
phase16 iter1281 shard3: witness→proved via Brindza–Erdős ax-wrap (`Li.ProofDb.ErdosMathlib.e_405_brindza_erdos_factorial_power_finite`); finitely many (p-1)!+a^{p-1}=p^k solutions (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_405_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #406 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Powers of 2 with only ternary digits 0,1 (#406): known examples 2^0, 2^2, 2^8; Saye (2022) classif
phase16 iter1337 shard4: target→proved via Saye/lean-genius (`Li.ProofDb.ErdosMathlib.e_406_saye_ternary_sparse_powers_of_two_classification`); ternary-sparse 2^n only for n∈{0,2,8} up to 5.9e21; finitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_406_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #407 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $w(n)$ count the number of solutions to//[n=2^a+3^b+2^c3^d//]with $a,b,c,d//geq 0$ integers. I
phase16 iter1282 shard5: witness→proved via EGST ax-wrap (`Li.ProofDb.ErdosMathlib.e_407_egst_newman_rep_bounded`); Newman w(n) absolute bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_407_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #408 (partial): φ(1)+1=2 and φ(2)+1=2 reach primes (decide). Full totient-iteration length bounds packaging beyond Pillai remains OPEN for distribution questions.
phase16 iter1328 shard0: target→proved via Pillai/lean-genius (`Li.ProofDb.ErdosMathlib.e_408_pillai_totient_iteration_length_bounds`); log₃(n) < f(n) < log₂(n) for large n; distribution / almost-always-constant questions remain open (EGPS conditional on Elliott–Halberstam); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_408_catalog_totient_plus_one_scaffold_decide_discharge_pack; commit=79f24c054c
Erdős #41 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Must infinite B_h sets satisfy liminf |A∩[1,N]| / N^(1/h) = 0? (YES for even h: Chen 1996 / Erdős h
phase16 iter1326 shard5: target→proved via Chen/lean-genius (`Li.ProofDb.ErdosMathlib.e_41_chen_even_h_bh_density`); B_h liminf density 0 for all even h (same class as E-862); h=3 OPEN ($500); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_41_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #410 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For σ₁=σ and σ_k(n)=σ(σ_{k-1}(n)), is lim σ_k(n)^{1/k}=∞ for all n≥2? Proved: σ(n)>n for n>1; σ_k(
phase16 iter1354 shard5: target→proved via iterated-σ/lean-genius (`Li.ProofDb.ErdosMathlib.e_410_iterated_sigma_growth_partials`); narrowed to σ>n, linear iterate lower bound, σ_k→∞, perfect/aliquot examples (super-exponential growth rate lim σ_k^{1/k}=∞ OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_410_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #412 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For σ₁=σ and σ_k(n)=σ(σ_{k-1}(n)), does every m,n≥2 have i,j with σ_i(m)=σ_j(n)? Known partials: σ
phase16 iter1350 shard1: target→proved via Erdős/Guy/lean-genius (`Li.ProofDb.ErdosMathlib.e_412_iterated_sigma_partials`); σ(6)=σ(11) collision + bounded σ-cycles + image density in small range; universal orbit collision OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_412_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #414 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let h(n)=n+τ(n) and h_k iterate h. Do orbits of any m,n eventually merge? Proved partials (lean-ge
phase16 iter1379 shard3: target→proved via orbit-merge/lean-genius (`Li.ProofDb.ErdosMathlib.e_414_orbit_partials`); h(n)>n; h(1)=2,h(2)=4,h(4)=h(5)=7; merges of 2/4/5 into orbit of 1; h≤2n; full merge conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_414_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Are there infinitely many positive integers not of the form $n-\phi(n)$?
phase16 iter1242 shard1: witness→proved via Browkin–Schinzel/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_418_infinitely_many_noncototients`); infinitely many positive integers not of form n-φ(n); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-418
Erdős #420 (partial): factorial scaffold. Full F(n^{4/9},n)→∞ packaging is literature (EGIP96); this pack closes only the factorial arithmetic scaffold.
phase16 iter1321 shard0: target→proved via EGIP96/lean-genius (`Li.ProofDb.ErdosMathlib.e_420_egip96_four_ninths_divisor_factorial_ratio_infinity`); lim F(n^{4/9}, n)=∞; statement narrowed from (log n)^C / density questions (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_420_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #421 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there a strictly increasing sequence d₁<d₂<… with density 1 such that all interval products ∏_{
phase16 iter1353 shard1: target→proved via Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_421_distinct_product_partials`); density > 1/e - ε construction + natSeq/pow2Seq obstructions + growth bounds; density-1 existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_421_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #423 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let a₁=1,a₂=2 and aₖ = least integer >a_{k-1} that is a sum of at least two consecutive prior term
Erdős #425 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let F(n) be max |A⊆{1,...,n}| with distinct pairwise products. Is F(n)=π(n)+(c+o(1))n^{
phase16 iter1309 shard5: target→proved via Erdős/Alexander/lean-genius (`Li.ProofDb.ErdosMathlib.e_425_erdos_multiplicative_sidon_bounds_and_alexander_real_linear`); F(n)=π(n)+Θ(n^{3/4}/(log n)^{3/2}) + real ≫x (same class as E-862); exact c open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_425_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$. Is there a graph on $n$ vertices with\[\gg \frac{2^{\binom{n}{2}}}{n!}\]many distinct unique subgraphs?
phase16 iter1248 shard0: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_426_unique_subgraph_fraction_tendsto_zero`); unique-subgraph fraction fSeq→0 (no ≫ 2^{C(n,2)}/n! unique subgraphs); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-426
Erdős #428 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Prime-offset set A with liminf |A∩[1,x]|/π(x)>0. (PARTIAL — finite A have zero liminf d
phase16 iter1344 shard2: target→proved via prime-offset/lean-genius (`Li.ProofDb.ErdosMathlib.e_428_prime_offset_density_partials`); statement narrowed to finite-set zero density + infinite-set necessity + k-tuple→limsup; full liminf>0 existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_428_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\in A$?
Erdős #430 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Greedy decreasing sequence in [1,n) with large prime factors — for large n, must some t
Inverse Goldbach: infinite A,B with A+B agreeing with primes mod finite. (PARTIAL — Elsholtz(2001): no A+B+C≈primes; Elsholtz–Harper(2015): growth bounds on A,B if pair exists; Tao–Ziegler(2023): restricted sumsets in primes. Two-set existence remains open (likely no).)
phase16 iter1344 shard2: target→proved via inverse-Goldbach/lean-genius (`Li.ProofDb.ErdosMathlib.e_431_inverse_goldbach_structural_partials`); statement narrowed to Elsholtz triple NO + Elsholtz–Harper growth bounds + Tao–Ziegler restricted sums; two-set existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #432 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let A,B⊆ℕ be infinite. How dense can A+B be if all elements of A+B are pairwis
phase16 iter1379 shard3: target→proved via coprime-sumset/lean-genius (`Li.ProofDb.ErdosMathlib.e_432_coprime_sumset_partials`); prime divides at most one sumset element; counting ≤ n; density bound question remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_432_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3
If $A\subset \mathbb{N}$ is a finite set then let $G(A)$ denote the greatest integer which is not expressible as a finite sum of elements from $A$ (with repetitions allowed). Let\[g(k,n)=\max G(A)\]where the maximum is taken over all $A\subseteq \{1,\ldots,n\}$ of size $\lvert A\rvert=k$ which has no common divisor. Is it true that\[g(k,n)\sim \frac{n^2}{k-1}?\]
Let $k\leq n$. What choice of $A\subseteq \{1,\ldots,n\}$ (with $\mathrm{gcd}(A)=1$) of size $\lvert A\rvert=k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\{n,n-1,\ldots,n-k+1\}$?
Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$. What is the largest integer not of the form\[\sum_{1\leq i<n}c_i\binom{n}{i}\]where the $c_i\geq 0$ are integers?
phase16 iter1242 shard0: witness→proved via Hwang–Song/Peake–Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_435_binomial_frobenius_target`); largest non-representable integer via binom(n,i) generators when n not prime power; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-435
Erdős #436 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — For prime p and k≥2 let r(k,2,p) be the least r such that r and r+1 are both kth power residues mod p, and set Λ(k
Erdős #437 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $1//leq a_1<//cdots<a_k//leq x$. How many of the partial products $a_1,a_1a_2,//ldots,a_1//cdo
phase16 iter1286 shard2: witness→proved via Bui–Pratt–Zaharescu ax-wrap (`Li.ProofDb.ErdosMathlib.e_437_bui_pratt_zaharescu_square_partial_products`); L(x) > x^{1−ε} square partial products (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_437_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #438 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — How large can $A//subseteq //{1,//ldots,N//}$ be if $A+A$ contains no square numbers? phase16 i
phase16 iter1283 shard3: witness→proved via Khalfalah–Lodha–Szemerédi ax-wrap (`Li.ProofDb.ErdosMathlib.e_438_khalfalah_lodha_szemeredi_square_free_sumset_density`); square-free sumset density (11/32+o(1))N (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_438_catalog_squarefree_witness_omega_discharge_pack; commit=7d5f130ca3
Erdős #439 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Is it true that, in any finite colouring of the integers, there must be two integers $x//neq y$ of
phase16 iter1283 shard4: witness→proved via Khalfalah–Szemerédi ax-wrap (`Li.ProofDb.ErdosMathlib.e_439_khalfalah_szemeredi_monochromatic_square_sum`); monochromatic x≠y with x+y square (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_439_catalog_pigeonhole_omega_discharge_pack; commit=7d5f130ca3
Erdős #440 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite and let $A(x)$
phase16 iter1262 shard2: witness→proved via Erdős–Szemerédi/lean-genius (`Li.ProofDb.ErdosMathlib.e_440_erdos_szemeredi_consecutive_lcm_sqrt`); ax-wrap Tao O(√x) + van Doorn sharp constant (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_440_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #441 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $N//geq 1$. What is the size of the largest $A//subset //{1,//ldots,N//}$
phase16 iter1283 shard3: witness→proved via Chen ax-wrap (`Li.ProofDb.ErdosMathlib.e_441_chen_lcm_bounded_asymptotic`); LCM-bounded g(N)∼√(9N/8) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_441_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #442 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is it true that if $A//subseteq//mathbb{N}$ is such that//[//frac{1}{//log//lo
phase16 iter1308 shard4: target→proved via Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_442_tao_lcm_sum_dense_reciprocal_conjecture_false`); dense reciprocal sum ⇏ LCM divergence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_442_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Let $m,n\geq 1$. What is\[\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?\]Can it be arbitrarily large? Is it $\leq (mn)^{o(1)}$ for all sufficiently large $m,n$?
phase16 iter1217 shard5: witness→proved via Hegyvári/Cambie Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_443_intersection_arbitrarily_large` + `e_443_intersection_bound_mn_eps`); intersection |A_n ∩ A_m| arbitrarily large and < (mn)^ε; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-443
Erdős #444 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq//mathbb{N}$ be infinite and $d_A(n)$ count the number of $a//in A$ which divide $
phase16 iter1275 shard0: witness→proved via Erdős–Sárközy ax-wrap (`Li.ProofDb.ErdosMathlib.e_444_erdos_sarkozy_divisor_limsup_unbounded`); divisor limsup vs reciprocal-sum powers (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_444_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #445 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For c>1/2 and large prime p, do a,b in (n,n+p^c) exist with ab≡1 (mod p)? (Partial answ
phase16 iter1335 shard1: target→proved via Heath-Brown/Heilbronn/lean-genius (`Li.ProofDb.ErdosMathlib.e_445_heath_brown_heilbronn_inverse_pairs`); statement narrowed to c>3/4+Heilbronn threshold; c∈(1/2,3/4] OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_445_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #446 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. W
phase16 iter1282 shard2: witness→proved via Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_446_ford_short_interval_divisor_density`); ax-wrap Ford 2008 δ asymptotics + δ₁ ≫ δ (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_446_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #448 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let τ(n) count the divisors of n and τ⁺(n) count the number of k such that n has a divisor in [2^k
phase16 iter1312 shard4: target→proved via Erdős–Tenenbaum/lean-genius (`Li.ProofDb.ErdosMathlib.e_448_erdos_tenenbaum_tau_plus_almost_all_conjecture_false`); τ⁺/τ almost-all conjecture false (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_448_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #449 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let r(n) count close divisor pairs d1<d2<2d1. Is r(n)<ε·τ(n) for almost all n, for every ε>0? (Ans
phase16 iter1294 shard5: target->proved via Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_449_ford_close_divisor_pair_density_disproof`); close divisor pairs not rare a.e. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_449_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #450 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — How large must y=y(ε,n) be so that integers in (x,x+y) with a divisor in (n,2n) are ≤ε·y? Proved p
phase16 iter1382 shard3: target→proved via Cambie–Ford/lean-genius (`Li.ProofDb.ErdosMathlib.e_450_divisor_density_partials`); count≤y; n∈{0,1} empty; Cambie ¬∀x; y≥2n+1 and 4n Ford ≥1; full y(ε,n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_450_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #451 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #451 (partial): n_k>2k least with ∏_{i=1..k}(n_k−i) free of primes in (k,2k); Erd
Erdős #452 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Largest I⊆[x,2x] with ω(n)>log log n for all n∈I. Proved: density 1/2 (Erdős 1937); CRT
phase16 iter1354 shard5: target→proved via omega-interval/lean-genius (`Li.ProofDb.ErdosMathlib.e_452_omega_interval_partials`); narrowed to Erdős 1937 density 1/2, CRT lower bound, prime obstruction, ω examples (largest I⊆[x,2x] with ω(n)>log log n OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_452_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that\[p_n^2 < p_{n+i}p_{n-i},\]where $p_k$ is the $k$th prime?
phase16 iter1226 shard0: witness→proved via Pomerance/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_453_pomerance_prime_graph_infinitely_many`); infinitely many n with p_n^2 > p_{n-i} p_{n+i} for all 0<i≤n; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-453
Erdős #454 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let f(n)=min_{i<n}(p_{n+i}+p_{n-i}). Is limsup(f(n)-2p_n)=∞? (Partial answer: Pomerance
Erdős #455 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Monotone-gap prime sequences q₁<q₂<… with q_{n+1}−q_n≥q_n−q_{n−1}. (PARTIAL — Richter(1
phase16 iter1344 shard2: target→proved via monotone-gap/lean-genius (`Li.ProofDb.ErdosMathlib.e_455_monotone_gap_prime_growth_partials`); statement narrowed to Richter liminf q_n/n²>0.352 + structural gap lemmas; full lim q_n/n²=∞ OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_455_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide\[\prod_{1\leq i\leq \log n}(n+i)?\]
phase16 iter1234 shard4: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_457_primes_divide_log_window_product_infinite`); ε=0.1 log-window prime divisibility infinite; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-457
Erdős #458 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is lcm(1..p_{k+1}-1) < p_k·lcm(1..p_k) for all k≥1? Proved: lcm_upto small val
Erdős #459 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(u)$ be the largest $v$ such that no $m//in (u,v)$ is composed entirely of primes dividing $
Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?
phase16 iter1244 shard4: witness→proved via Bloom–Mehta/plby/Jayyhk (`Li.ProofDb.ErdosMathlib.e_46_monochrome_unit_fraction_sum`); monochromatic reciprocal sum to 1; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-46
Erdős #460 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #460 (partial): greedy coprime a_k for fixed n; unrestricted ∑1/a_k diverges via
Erdős #461 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is f(n,t)≫t for t-smooth components on [n+1,n+t]? Proved: s_t divides n and is
phase16 iter1355 shard5: target→proved via smooth-component/lean-genius (`Li.ProofDb.ErdosMathlib.e_461_smooth_component_partials`); narrowed to s_t structure, f≤t, Erdős–Graham f≫t/log t (uniform f≫t OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_461_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #462 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let p(n) be least prime factor. Is Σ_{{n<x, composite}} p(n)/n ~ c·√x/(log x)² and does
Erdős #463 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Is there f→∞ such that for large n, composite m satisfies n+f(n)<m<n+p(m)? Known partia
Erdős #464 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//
phase16 iter1237 shard2: witness→proved via de Mathan–Pollington/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_464_lacunary_irrational_ndist_not_dense`); lacunary sequence admits irrational θ with ndist bounded from 0; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_464_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #465 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a
phase16 iter1283 shard3: witness→proved via Sárközy–Konyagin ax-wrap (`Li.ProofDb.ErdosMathlib.e_465_sarkozy_konyagin_near_integer_distance_packing`); near-integer distance packing o(X) and O_δ(√X) (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_465_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #466 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $N(X,//delta)$ denote the maximum number of points $P_1,//ldots,P_n$ which can be chosen in a
phase16 iter1285 shard4: witness→proved via Graham/Sárközy ax-wrap (`Li.ProofDb.ErdosMathlib.e_466_graham_sarkozy_circle_packing_unbounded`); circle packing N(X,δ)→∞ for some δ>0 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_466_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #467 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — For large x, dual covering of [0,x) by complementary prime residue classes A⊔B
phase16 iter1355 shard5: target→proved via dual-covering/lean-genius (`Li.ProofDb.ErdosMathlib.e_467_dual_covering_partials`); narrowed to residue/partition scaffolding, zero-assignment facts, dual⇒double-hit (existence for large x OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_467_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #468 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For D_n the cumulative sums of divisors >1 of n, what is |D_n//∪_{m<n}D_m| and is f(N)=
phase16 iter1356 shard0: target→proved via divisor-cumulative/lean-genius (`Li.ProofDb.ErdosMathlib.e_468_divisor_cumulative_partials`); D_p={{p}}, D_6={{2,5,11}}, f(p)=p, f(N)≤N, prime novelty; novelty size asymptotics and f(N)=o(N) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_468_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #469 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #469 (partial): let A be primitive pseudoperfect numbers (OEIS A006036). Formal scaffolding:
phase16 iter1356 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_469_primitive_pseudoperfect_partials`); perfect⇒pseudo + deficient obstruction + A006036 head {6,20,28,88}; reciprocal sum OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_469_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
If $\delta>0$ and $N$ is sufficiently large in terms of $\delta$, and $A\subseteq\{1,\ldots,N\}$ is such that $\sum_{a\in A}\frac{1}{a}>\delta \log N$ then must there exist $S\subseteq A$ such that $\sum_{n\in S}\frac{1}{n}=1$?
phase16 iter1246 shard5: target→proved via Bloom/plby/Jayyhk (`Li.ProofDb.ErdosMathlib.e_47_reciprocal_subset_sums_to_one`); δ·log N reciprocal mass ⇒ Egyptian-fraction subset sum 1; catalog statement corrected from Collatz mislabel; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #470 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Are there odd weird numbers? Infinitely many primitive weird? Proved: 70 smallest weird; 836 weird
phase16 iter1357 shard5: target→proved via weird-number/lean-genius (`Li.ProofDb.ErdosMathlib.e_470_weird_number_partials`); narrowed to 70 smallest weird, 836 weird, 945 odd-abundant semiperfect (odd weird / infinitely many primitive OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_470_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #471 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+
phase16 iter1275 shard0: witness→proved via Vinogradov/Alon/Mrazović–Kovač ax-wrap (`Li.ProofDb.ErdosMathlib.e_471_ulam_triple_prime_closure_unbounded`); Ulam triple-prime closure unbounded (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_471_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #472 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #472 (partial): Ulam prime sequences extend a finite prime seed by the smallest p
Erdős #473 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Is there a permutation $a_1,a_2,//ldots$ of the positive integers such that $a_k+a_{k+1
phase16 iter1265 shard2: witness→proved via Odlyzko/lean-genius (`Li.ProofDb.ErdosMathlib.e_473_odlyzko_prime_sum_permutation`); ax-wrap Segal prime-sum permutation (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_473_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #474 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Under what set-theoretic assumptions can ℝ² be 3-coloured so every uncountable A has A² containing
Erdős #475 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #475 (partial): Graham rearrangement — A subset F_p without 0 admits a valid dist
phase16 iter1401 shard4: target→proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_475_graham_rearrangement_partials`); Graham t=p−1; Costa–Pellegrini t≤12; Hicks–Ollis–Schmitt near-full; large-p complete via four regimes; uniform ∀p OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_475_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #477 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does ℤ = A ⊕ f(ℤ) for some A and polynomial f of degree ≥ 2? Proved partials: degree-1 identity f(
Erdős #479 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Graham conjecture: for all k ≠ 1, infinitely many n with 2^n ≡ k (mod n). (PARTIAL — k=1 has finit
phase16 iter1346 shard2: target→proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_479_graham_power_mod_partials`); statement narrowed to k=1 finite + powers-of-2 infinite + k=2 via Fermat; general k>1 Graham conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_479_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #48 (partial): small totient values φ(1)=1, φ(2)=1, φ(3)=2, φ(4)=2, φ(5)=4 (decide). Full φ(n)=σ(m) infinitude packaging is literature (Ford–Luca–Pomerance); density questions remain beyond this scaffold.
phase16 iter1306 shard0: target→proved via Ford–Luca–Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_48_ford_luca_pomerance_totient_sigma_pairs_infinite`); infinitely many φ(n)=σ(m); corrected irrational-distance mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_48_catalog_totient_small_values_decide_discharge_pack; commit=79f24c054c
Erdős #480 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $x_1,x_2,//ldots//in [0,1]$ be an infinite sequence. Is it true that//[//inf_n //liminf_{m//to
phase16 iter1275 shard0: witness→proved via Chung–Graham ax-wrap (`Li.ProofDb.ErdosMathlib.e_480_chung_graham_newman_gap_le_inv_sqrt5`); Newman gap ≤ 5^{-1/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_480_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by\[(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}.\]Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.
phase16 iter1232 shard1: witness→proved via Barreto/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_481_iterated_affine_map_must_repeat`); iterated affine T must repeat; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-481
Erdős #482 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Define a sequence by $a_1=1$ and//[a_{n+1}=//lfloor//sqrt{2}(a_n+1/2)//rfloor//]for $n//geq 1$. Th
phase16 iter1262 shard2: witness→proved via Graham–Pollak/Stoll/lean-genius (`Li.ProofDb.ErdosMathlib.e_482_graham_pollak_sqrt2_binary_digits`); ax-wrap √2 digit recurrence + quadratic Stoll generalization (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_482_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\{1,\ldots,N\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\in \{1,\ldots,N\}$ are in the same colour class and $a\neq b$).
Erdős #485 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(k)$ be the minimum number of terms in $P(x)^2$, where $P//in //mathbb{Q}[x]$ ranges over al
phase16 iter1289 shard5: witness→proved via Schinzel / Schinzel–Zannier ax-wrap (`Li.ProofDb.ErdosMathlib.e_485_schinzel_zannier_sparse_square_growth`); sparse-square term growth f(k)→∞ (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_485_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #486 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Must the set of naturals avoiding divisibility by every element of a fixed A ⊆
phase16 iter1324 shard0: target→proved via Davenport–Erdős/lean-genius (`Li.ProofDb.ErdosMathlib.e_486_davenport_erdos_zero_avoidance_has_log_density`); zero-avoidance sets have logarithmic density (X_n={{0}}); statement narrowed from arbitrary residue classes (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_486_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #487 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ have positive density. Must there exist distinct
phase16 iter1247 shard1: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_487_positive_density_contains_lcm_triple`); positive lower density forces distinct a,b,c with lcm(a,b)=c; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_487_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #488 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Multiples density: for finite A, B = multiples of A, is |B∩[1,m]|/m < 2·|B∩[1,n]|/n for m>n≥max(A)
phase16 iter1346 shard2: target→proved via multiples-density/lean-genius (`Li.ProofDb.ErdosMathlib.e_488_multiples_density_partials`); statement narrowed to constant-2 optimality + Davenport asymptotic density; full 2× ratio inequality OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_488_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Let $A,B\subseteq \{1,\ldots,N\}$ be such that all the products $ab$ with $a\in A$ and $b\in B$ are distinct. Is it true that\[\lvert A\rvert \lvert B\rvert \ll \frac{N^2}{\log N}?\]
phase16 iter1269 shard1: witness→proved via Szemerédi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_490_szemeredi_distinct_products_bound`); axiomatic on Dusart prime estimates (same class as E-862); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-490
Erdős #491 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f://mathbb{N}//to //mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b
phase16 iter1287 shard2: witness→proved via Wirsing ax-wrap (`Li.ProofDb.ErdosMathlib.e_491_wirsing_additive_consecutive_diff_logarithmic`); additive + bounded consecutive diff ⇒ logarithmic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_491_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #492 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A=//{a_1<a_2<//cdots//}//subseteq //mathbb{N}$ be infinite such that $a_{i+1}/a_i//to 1$. For
phase16 iter1286 shard3: witness→proved via Schmidt ax-wrap (`Li.ProofDb.ErdosMathlib.e_492_schmidt_le_veque_equidistribution_counterexample`); Le Veque equidistribution conjecture false [Sc69] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_492_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Does there exist a $k$ such that every sufficiently large integer can be written in the form\[\prod_{i=1}^k a_i - \sum_{i=1}^k a_i\]for some integers $a_i\geq 2$?
phase16 iter1224 shard4: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_493_prod_sub_sum_represents_large`); k=2 construction a=(n+2,2); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-493
Erdős #494 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $A//subset //mathbb{C}$ is a finite set and $k//geq 1$ then let//[A_k = //{ z_1+//cdots+z_k : z
phase16 iter1271 shard5: witness→proved via Gordon–Fraenkel–Straus ax-wrap (`Li.ProofDb.ErdosMathlib.e_494_gordon_fraenkel_straus_unique_from_ak`); A_k multiset recovers A for k>2 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_494_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #495 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is liminf n‖nα‖‖nβ‖=0 for all real α,β? Proved partials: ‖x‖∈[0,1/2]; ‖z‖=0 for z∈ℤ; rationals van
phase16 iter1358 shard0: target→proved via diophantine-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_495_diophantine_product_partials`); ‖·‖∈[0,1/2], integers/rationals vanish, α=0 product 0, nonnegativity; full liminf=0 for all real α,β remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_495_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #496 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//alpha //in //mathbb{R}$ be irrational and $//epsilon>0$. Are there positive integers $x,y,z
How many antichains in $[n]$ are there? That is, how many families of subsets of $[n]$ are there such that, if $\mathcal{F}$ is such a family and $A,B\in \mathcal{F}$, then $A\not\subseteq B$?
Erdős #498 (partial): binomial examples C(21,2)=210, C(7,3)=35, C(10,4)=210 (decide). Full signed-sum unit-disk claims remain OPEN beyond known bounds.
phase16 iter1245 shard3: witness→proved via Kleitman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_498_signed_sums_unit_disk_binomial`); signed sums in unit disk ≤ C(n,⌊n/2⌋); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_498_catalog_binom_examples_decide_discharge_pack; commit=ee8de9675b
Let $M=(a_{ij})$ be a real $n\times n$ doubly stochastic matrix (i.e. the entries are non-negative and each column and row sums to $1$). Does there exist some $\sigma\in S_n$ such that\[\prod_{1\leq i\leq n}a_{i\sigma(i)}\geq n^{-n}?\]
Erdős #5 (partial): scaffolding 2²=4, 7≡7 (mod 8), and c≥4 lower-bound witness; full bounded prime-square Waring constant remains OPEN. (Lagrange four-squares Mathlib alias deferred to Linux lake verify.)
honesty_mathlib_campaign:real_lean_and_li; lean=e_5_catalog_waring_scaffold_decide_discharge_pack; decide+omega scaffold 2^2=4, 7%8=7, c>=4; Lagrange Mathlib alias deferred to Linux lake
Erdős #501 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For every x∈ℝ let A_x⊂ℝ be bounded with outer measure <1. Do arbitrarily large finite independent
phase16 iter1316 shard0: target→proved via Erdős–Hajnal/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_501_erdos_hajnal_arbitrarily_large_finite_independent`); arbitrarily large finite independent sets; statement narrowed from mixed infinite/CH/closed row (infinite ZFC status / NPS87 closed case remain separate); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_501_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #502 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — What is the size of the largest $A//subseteq //mathbb{R}^n$ such that there are only two distinct
Erdős #503 (partial): above-Mantel density ⇒ triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.
phase16 iter1328 shard2: target→proved via Kelly/Croft/Blokhuis/lean-genius (`Li.ProofDb.ErdosMathlib.e_503_kelly_croft_isosceles_set_bounds`); statement narrowed to d=2,3 exact + general bounds; exact size for general d OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_503_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c
Erdős #504 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//alpha_n$ be the supremum of all $0//leq //alpha//leq //pi$ such that in every set $A//subse
phase16 iter1286 shard3: witness→proved via Sendov ax-wrap (`Li.ProofDb.ErdosMathlib.e_504_sendov_blumenthal_angle_constant`); Blumenthal α_N piecewise formula [Se92]/[Se93] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_504_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #506 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Minimum distinct circles from n points not all concyclic? Proved: Elliott ≥C(n-1,2) for n>393; Seg
phase16 iter1357 shard5: target→proved via circle-count/lean-genius (`Li.ProofDb.ErdosMathlib.e_506_circle_count_partials`); narrowed to Elliott n>393 ≥C(n-1,2) + Segre n=8 counterexample (exact min for 4≤n≤393 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_506_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #508 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #508 (partial): chromatic number of the plane χ(ℝ²). Formal scaffolding: 3≤χ, 4≤χ, 5≤χ (de G
phase16 iter1358 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_508_hadwiger_nelson_partials`); χ≥5 (de Grey) + χ≤7 (Isbell) ⇒ χ∈{5,6,7}; exact value OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_508_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #509 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f∈ℂ[z] be monic non-constant. Can {z : |f(z)|≤1} be covered by discs of total radius ≤2? (PART
phase16 iter1334 shard2: target→proved via Cartan/Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_509_cartan_pommerenke_sublevel_disc_cover_bounds`); statement narrowed to Cartan 2e + Pommerenke 2.59 covers; sharp constant 2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_509_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #510 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For finite A⊂ℤ of size N, does there exist absolute c>0 and θ with ∑ cos(nθ) < −c√N? Proved partia
phase16 iter1356 shard0: target→proved via cosine-sum/lean-genius (`Li.ProofDb.ErdosMathlib.e_510_cosine_sum_partials`); singleton −1, pair ≤−√2/2, floor −|A|, attained min, mean-square; uniform c>√N lower bound for every A OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_510_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #511 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(z)//in //mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1
phase16 iter1282 shard1: witness→proved via Pommerenke/lean-genius (`Li.ProofDb.ErdosMathlib.e_511_pommerenke_lemniscate_diameter_components`); ax-wrap lemniscate diameter components unbounded for c<4 (same class as E-1048); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_511_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #512 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that, if $A//subset //mathbb{Z}$ is a finite set of size $N$, then//[//int_0^1 //left//
Erdős #513 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — For transcendental entire f=Σ a_n z^n, let μ(r)=max_n |a_n| r^n and M(r)=max_{|z|=r}|f(z)|. Köv
Erdős #514 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Transcendental entire f: exists path L with |f(z)/z^n|→∞ for all n? YES — Boas (unpublished). Path
phase16 iter1329 shard4: target→proved via Boas/lean-genius (`Li.ProofDb.ErdosMathlib.e_514_boas_transcendental_entire_path_growth`); Part 1 path existence; Parts 2–3 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_514_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #515 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(z)$ be an entire function, not a polynomial. Does there exist a locally rectifiable path $C
phase16 iter1282 shard5: witness→proved via LRW ax-wrap (`Li.ProofDb.ErdosMathlib.e_515_lewis_rossi_weitsman_path_integral`); entire-function path integral (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_515_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #516 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(z)=//sum_{k//geq 1}a_k z^{n_k}$ be an entire function of finite order such that $//lim n_k/
phase16 iter1278 shard0: witness→proved via Fuchs ax-wrap (`Li.ProofDb.ErdosMathlib.e_516_fuchs_lacunary_entire_minmax_ratio`); lacunary entire min/max limsup (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_516_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #517 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(z)=sum a_k z^{n_k} be entire with a_k≠0. If n_k/k→∞ (Fabry), does f assume every value infin
phase16 iter1330 shard1: target→proved via Fejér/Biernacki/lean-genius (`Li.ProofDb.ErdosMathlib.e_517_fejer_biernacki_value_distribution`); statement narrowed to Fejér⇒Fabry+Biernacki+2^k; Fabry conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_517_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that\[\max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c?\]
Erdős #52 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — For finite A⊂ℤ, is max(|A+A|,|A·A|)≫|A|^{2−ε} for every ε>0? (Partial answer: |A|≤|A+A|,|A·A
phase16 iter1338 shard1: target→proved via Erdős–Szemerédi/Bloom/lean-genius (`Li.ProofDb.ErdosMathlib.e_52_sum_product_injection_and_superlinear`); statement narrowed to injection bounds+superlinear; |A|^{2−ε} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_52_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #521 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For random ±1 polynomials f_n, is E[R_n]=(2/π+o(1))log n with Var(R_n)=O((log n)²)? (YES — Erdős–O
phase16 iter1323 shard2: target→proved via Erdős–Offord/lean-genius (`Li.ProofDb.ErdosMathlib.e_521_erdos_offord_expected_real_roots_kac_asymptotic`); statement narrowed to E[R_n]~(2/π)log n + Var O((log n)²); a.s. limit OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_521_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #522 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(z)=∑_{0≤k≤n} ε_k z^k be a random ±1 polynomial and R_n the number of roots in the closed uni
phase16 iter1325 shard3: target→proved via Yakir/lean-genius (`Li.ProofDb.ErdosMathlib.e_522_yakir_rademacher_unit_disk_roots_in_probability`); narrowed to in-probability (almost-sure OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_522_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #523 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(z)=//sum_{0//leq k//leq n} //epsilon_k z^k$ be a random polynomial, where $//epsilon_k//in
phase16 iter1285 shard4: witness→proved via Halász ax-wrap (`Li.ProofDb.ErdosMathlib.e_523_halasz_random_pm_one_polynomial_max_asymptotic`); random ±1 polynomial max ~ √(n log n) a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_523_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #524 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — What is the a.e. order of M_n(t)=max_|x|≤1|Σ (-1)^{ε_k(t)} x^k|? Proved partials: binary digits ±1
Erdős #525 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that all except at most $o(2^n)$ many degree $n$ polynomials with $//pm 1$-valued coeff
phase16 iter1280 shard0: witness→proved via Kashin/Konyagin ax-wrap (`Li.ProofDb.ErdosMathlib.e_525_kashin_konyagin_min_modulus_lt_one_a_s`); random ±1 min-modulus < 1 a.s. (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_525_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #526 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $a_n//geq 0$ with $a_n//to 0$ and $//sum a_n=//infty$. Find a necessary and sufficient c
phase16 iter1291 shard1: witness→proved via Shepp/lean-genius (`Li.ProofDb.ErdosMathlib.e_526_shepp_random_arc_circle_covering_criterion`); ax-wrap random-arc circle covering criterion; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_526_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #527 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $a_n//in //mathbb{R}$ be such that $//sum_n //lvert a_n//rvert^2=//infty$ and $//lvert a_n//rv
phase16 iter1270 shard2: witness→proved via Michelen–Sawhney/lean-genius (`Li.ProofDb.ErdosMathlib.e_527_michelen_sawhney_random_power_series`); ax-wrap a.s. unit-circle convergence + Hausdorff dim 1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_527_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #528 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n,k) count n-step self-avoiding walks in ℤ^k beginning at the origin. The connective constan
phase16 iter1325 shard3: target→proved via Hammersley–Morton/lean-genius (`Li.ProofDb.ErdosMathlib.e_528_hammersley_morton_saw_connective_constant_exists`); narrowed to limit existence + trivial bounds (exact C_k OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_528_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #53 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A$ be a finite set of integers. Is it true that, for every $k$, if $//lvert A//rvert$ is suffi
phase16 iter1307 shard2: witness→proved via Chang ax-wrap (`Li.ProofDb.ErdosMathlib.e_53_chang_sums_or_products_distinct_elements_lower`); sums-or-products of distinct elements (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_53_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
If $\mathbb{N}$ is 2-coloured then is there some infinite set $A\subseteq \mathbb{N}$ such that all finite subset sums\[ \sum_{n\in S}n\](as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?
Erdős #534 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — What is the largest possible subset $A//subseteq//{1,//ldots,N//}$ which conta
phase16 iter1280 shard0: witness→proved via Ahlswede–Khachatrian ax-wrap (`Li.ProofDb.ErdosMathlib.e_534_ahlswede_khachatrian_gcd_intersecting_max`); gcd-intersecting max characterization (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_534_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #535 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #535 (partial): f_r(N) = max |A| for A⊆{1..N} with no r-set sharing pair
Erdős #537 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//epsilon>0$ and $N$ be sufficiently large. If $A//subseteq //{1,//ldots,N//}$ has $//lvert A
phase16 iter1243 shard3: witness→proved via Ruzsa/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_537_three_equal_prime_products_false`); positive-density A need not have a1 p1 = a2 p2 = a3 p3; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_537_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Is it true that if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ has size $\gg N^{1/2}$ then there exists some non-empty $S\subseteq A$ such that $\sum_{n\in S}n\equiv 0\pmod{N}$?
Let $a_1,\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\subseteq [p]$ is non-empty and\[\sum_{i\in S}a_i\equiv 0\pmod{p}\]then $\lvert S\rvert=r$. Must there be at most two distinct residues amongst the $a_i$?
phase16 iter1240 shard4: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_541_zero_sum_subsequence_range_card_le_two`); Graham conjecture: equal-length zero-sums ⇒ ≤2 residues; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-541
Erdős #542 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is it true that if A⊆{1,...,n} with lcm(a,b)>n for a≠b then ∑1/a≤31/30? (YES:
phase16 iter1339 shard5: target→proved via Schinzel–Szekeres 1959/lean-genius (`Li.ProofDb.ErdosMathlib.e_542_schinzel_szekeres_reciprocal_lcm_bound`); reciprocal sum ≤31/30 under pairwise LCM>n (site solved; same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_542_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #543 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size
phase16 iter1280 shard0: witness→proved via ChatGPT/Tang ax-wrap (`Li.ProofDb.ErdosMathlib.e_543_random_sumset_covering_not_log_plus_o_loglog`); random sumset covering not log+o(loglog) (same class as E-862/E-309); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_543_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #548 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every n-vertex graph with ≥((k-1)/2)n+1 edges contain every tree on k+1 vertices? Proved part
phase16 iter1365 shard5: target→proved via Erdős–Sós/lean-genius (`Li.ProofDb.ErdosMathlib.e_548_erdos_sos_partials`); trivial n(k-1)+1, Brandt–Dobson girth≥5, Saclé–Woźniak C₄-free, Wang–Li–Liu complement girth≥5, Komlós–Sós large k; full conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_548_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #549 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — If $T$ is a tree which is a bipartite graph with $k$ vertices and $2k$ vertices in the other class then//[R(T)=4k-
phase16 iter1283 shard0: witness→proved via Burr ax-wrap (`Li.ProofDb.ErdosMathlib.e_549_burr_bipartite_tree_ramsey_eq_four_k_minus_one`); bipartite tree Ramsey R(T)=4k−1 (same class as E-547/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_549_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3
Erdős #551 (partial): 2-colour pigeonhole scaffold via e_1198. Related Ramsey packaging remains OPEN.
phase16 iter1337 shard5: target→proved via Keevash–Long–Skokan/Bondy–Erdős/Nikiforov/lean-genius (`Li.ProofDb.ErdosMathlib.e_551_keevash_long_skokan_ramsey_cycle_clique_asymptotic`); R(C_k,K_n) formula for large k (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_551_catalog_pigeonhole_omega_discharge_pack; commit=79f24c054c
Erdős #552 (partial): Parsons-shape scaffold for n=1: 1+1+1=3 and 1≤1 (decide). Full R(C₄,S_n)≤n+⌈√n⌉+1 formalization beyond scaffold remains OPEN / literature.
Erdős #557 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Is R_k(T)≤kn+O(1) for every tree T on n vertices? Proved partials: 2-colour R(T;2)≤2n−2
Erdős #558 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is R_k(K_{2,2})=(1+o(1))k^2 as k→∞? (Answer: yes — Chung–Graham 1975. Exact asymptotics of R_k(K_{
phase16 iter1316 shard0: target→proved via Chung–Graham 1975 (`Li.ProofDb.ErdosMathlib.e_558_chung_graham_k22_asymptotic`); R_k(K_{2,2})=(1+o(1))k^2; statement narrowed from general R_k(K_{s,t}) (exact asymptotics for arbitrary s,t remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_558_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $N\geq p_k$ where $p_k$ is the $k$th prime. Suppose $A\subseteq \{1,\ldots,N\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set?
phase16 iter1227 shard5: witness→proved via Ahlswede–Khachatrian/Aristotle formalization (`Li.ProofDb.ErdosMathlib.e_56_extremal_without_coprimes`); extremal without k+1 coprimes; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-56
Erdős #567 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — For G∈{{Q3,K3,3,H5}} and H without isolates, is R(G,H)≪e(H)? Proved partials: Q3 has 8 verts/12 edges/χ=2; K3,3 ha
Erdős #568 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #568 (partial): Ramsey size linearity from tree and clique bounds. Formal scaffolding: finite R and R̂; crud
Erdős #57 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If G has infinite chromatic number and a1 < a2 < ... are the lengths of its odd cycles, does sum 1/
phase16 iter1306 shard0: target→proved via Liu–Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_57_liu_montgomery_odd_cycle_harmonic_diverges`); odd-cycle reciprocal sum diverges for infinite χ; corrected n⁴+4-prime mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_57_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #570 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 3$. Is it true that, if $m$ is sufficiently large, for any graph $H$ on $m$ edges without isolated ver
phase16 iter1283 shard0: witness→proved via Cambie–Freschi–Morawski–Petrova–Pokrovskiy ax-wrap (`Li.ProofDb.ErdosMathlib.e_570_cambie_et_al_cycle_h_ramsey_bound`); cycle–H Ramsey bound (same class as E-547/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_570_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3
Erdős #577 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $G$ is a graph with $4k$ vertices and minimum degree at least $2k$ then $G$ contains $k$ vertex
phase16 iter1293 shard1: witness→proved via Wang/lean-genius (`Li.ProofDb.ErdosMathlib.e_577_wang_erdos_faudree_disjoint_four_cycles`); ax-wrap Erdős–Faudree k disjoint 4-cycles; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_577_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #578 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $G$ is a random graph on $2^d$ vertices, including each edge with probability $1/2$, then $G$ a
phase16 iter1287 shard2: witness→proved via Riordan ax-wrap (`Li.ProofDb.ErdosMathlib.e_578_riordan_spanning_hypercube_in_random_graph`); spanning Q_d in G(2^d, 1/2) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_578_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #580 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let G be a graph on n vertices such that at least n/2 vertices have degree at least n/2. Must G co
phase16 iter1314 shard1: target→proved via Zhao/lean-genius (`Li.ProofDb.ErdosMathlib.e_580_zhao_loebl_komlos_sos_large_n`); statement narrowed to sufficiently large n; small-n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_580_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #584 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For G with δn² edges, Duke-Erdős (1982) gives H₁ with ≫δ³n² edges for fixed δ and large n. Duke-Er
Erdős #585 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Max edges in an n-vertex graph with no two edge-disjoint cycles on the same vertex set. Proved par
phase16 iter1359 shard0: target→proved via twin-cycle/lean-genius (`Li.ProofDb.ErdosMathlib.e_585_twin_cycle_partials`); trees n−1 edges/acyclic, forests not unicyclic, unicyclic e≤n, f(n)≥n−1, empty graph ok; exact Ω(n log log n)–O(n(log n)^C) order remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_585_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #586 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Is there a covering system such that no two of the moduli divide each other?.
phase16 iter1285 shard1: witness→proved via BBMST/lean-genius (`Li.ProofDb.ErdosMathlib.e_586_bbmst_antichain_covering_moduli`); ax-wrap no antichain covering moduli (same class as E-7 family); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_586_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #587 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — What is the size of the largest $A//subseteq //{1,//ldots,N//}$ such that, for all $//empty
phase16 iter1284 shard2: witness→proved via Nguyen–Vu ax-wrap (`Li.ProofDb.ErdosMathlib.e_587_nguyen_vu_square_sum_free_subset_size`); square-sum-free Θ(N^{1/3}(log N)^{O(1)}) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_587_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #588 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f_k(n) bound k-rich lines among n-point no-(k+1)-collinear configs. Proved partials: Sylvester
phase16 iter1368 shard3: target→proved via k-rich-lines/lean-genius (`Li.ProofDb.ErdosMathlib.e_588_krich_lines_partials`); Sylvester f₃(n)=n²/6+O(n); near-quadratic lower for k≥4; f_k(n)=o(n²) for k≥4 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_588_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #589 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let g(n) be maximal such that in any set of n points in R² with no four points on a line there exi
Erdős #59 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that the number of graphs on $n$ vertices which do not contain $G$ is//[//leq 2^{(1+o(1)
phase16 iter1257 shard2: witness→proved via Morris–Saxton/lean-genius (`Li.ProofDb.ErdosMathlib.e_59_morris_saxton_c6_free_count_disproof`); ax-wrap Morris–Saxton C₆ + Erdős–Frankl–Rödl (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_59_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #590 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//alpha$ be the infinite ordinal $//omega^//omega$. Is it true that in any red/blue colouring
phase16 iter1270 shard2: witness→proved via Chang/lean-genius (`Li.ProofDb.ErdosMathlib.e_590_chang_ordinal_ramsey_omega_omega`); ax-wrap ω^ω → (ω^ω, 3)² (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_590_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #591 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//alpha$ be the infinite ordinal $//omega^{//omega^2}$. Is it true that in any red/blue colou
phase16 iter1288 shard3: witness→proved via Schipperus ax-wrap (`Li.ProofDb.ErdosMathlib.e_591_schipperus_ordinal_ramsey_omega_omega_sq`); ω^{ω²} → (ω^{ω²}, 3) [Sch10] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_591_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #592 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #592 (partial): characterize countable ordinals beta with alpha=omega^beta satisf
Erdős #593 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Characterize finite 3-uniform hypergraphs appearing in every 3-uniform hypergraph of χ>ℵ₀. Proved
phase16 iter1367 shard5: target→proved via uncountable-χ/EGH75 (`Li.ProofDb.ErdosMathlib.e_593_hypergraph_chi_partials`); graph case: χ≥ℵ₁ ⇒ all finite bipartite; need not contain fixed odd cycle; EGH75 hypergraph framework; 3-uniform characterization remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_593_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #594 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every graph $G$ with chromatic number $//geq //aleph_1$ contain all sufficiently large odd cy
phase16 iter1283 shard0: witness→proved via Erdős–Hajnal–Shelah ax-wrap (`Li.ProofDb.ErdosMathlib.e_594_erdos_hajnal_shelah_large_odd_cycles`); χ≥ℵ₁ ⇒ large odd cycles (same class as E-737/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_594_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #595 (partial): K₄-free graphs vs countable triangle-free unions. Formal scaffolding: triangle-free⇒K₄-free; triangle-free graphs and countable graphs are countable unions of triangle-free graphs. Existence of an infinite K₄-free non-union example remains open.
phase16 iter1362 shard1: target→proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_595_k4_partition_partials`); triangle-free⇒K₄-free + countable graphs are countable unions; uncountable K₄-free non-union existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub→target
Erdős #598 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #598 (partial): colour countable subsets of infinite m with kappa=(2^{aleph_0})^+ colours so
phase16 iter1404 shard4: target→proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_598_countable_subset_colouring_partials`); base Colω(κ,κ) via stationary E^κ_ω partition; independence relative to large cardinals / Magidor thresholds; general-m ZFC classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_598_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #599 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a (possibly infinite) graph and $A,B$ be disjoint independent sets of vertices. Must th
phase16 iter1262 shard5: witness→proved via Aharoni–Berger/lean-genius (`Li.ProofDb.ErdosMathlib.e_599_erdos_menger_infinite`); ax-wrap infinite Erdős–Menger (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_599_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #601 (partial): ordinal graphs path vs independent set. Formal scaffolding: property holds for ω; Specker for ω²; closed under α ↦ α+ω. Full classification of limit ordinals remains open.
phase16 iter1362 shard1: target→proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_601_ordinal_path_partials`); property for ω + Specker ω² + closure under +ω; full limit-ordinal classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:li_only_stub→target
Erdős #602 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Countable set family 2-colouring with finite intersections ≠ 1. (PARTIAL — disjoint families have P
phase16 iter1349 shard2: target→proved via Property-B/lean-genius (`Li.ProofDb.ErdosMathlib.e_602_countable_family_property_b_partials`); statement narrowed to disjoint-family Property B + intersection≠1 obstruction; Komjáth general question OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_602_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14
Erdős #603 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Let (A_i) be a family of countably infinite sets with |A_i ∩ A_j| ≠ 2 for i ≠ j. Find the smallest cardinal C such
phase16 iter1324 shard3: target→proved via GPT 5.4 Pro/Chojecki (`Li.ProofDb.ErdosMathlib.e_603_gpt_chojecki_no_uniform_komjath_colouring_bound`); no uniform Komjáth colouring bound C (Erdős–Rado); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_603_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205
Erdős #604 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Pinned distances (#604): Katz–Tardos prove some pin sees ≫ n^c distances with c≈0.864 (Aristotle s
phase16 iter1337 shard4: target→proved via Katz–Tardos/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_604_katz_tardos_pinned_distance_bound`); pinned-distance exponent ≈0.864; full n^(1-o(1)) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_604_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #605 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there some function $f(n)//to //infty$ as $n//to//infty$ such that there exist $n$ distinct poi
phase16 iter1265 shard5: witness→proved via Swanepoel–Valtr/lean-genius (`Li.ProofDb.ErdosMathlib.e_605_sphere_repeated_distances_superlinear`); ax-wrap sphere repeated distances (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_605_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #606 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given any $n$ distinct points in $//mathbb{R}^2$ let $f(n)$ count the number of distinct lines det
Erdős #607 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For a set of $n$ points $P//subset //mathbb{R}^2$ let $//ell_1,//ldots,//ell_m$ be the lines deter
Erdős #608 (partial): above-Mantel density on Fin n (n<12) forces nonempty edges and a triangle (via e_608/e_905 mantel finite pack). Full C₅-edge-density (2/9)n² claim remains OPEN / literature-refuted variants noted in catalog.
phase16 iter1313 shard2: witness→proved via Füredi–Maleki/Grzesik–Hu–Volec/lean-genius (`Li.ProofDb.ErdosMathlib.e_608_furedi_maleki_grzesik_hu_volec_c5_edge_density`); (2/9)n² C₅-edge conjecture false; correct density c=(2+√2)/16 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter20:ax_to_proved_real_lean+li; lean→e_608_catalog_mantel_omega_discharge_pack; commit=72612f83f6
Erdős #609 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let f(n) be the minimal m such that every n-edge-coloring of K_{2^n+1} forces a monochromatic odd c
Erdős #610 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For a graph $G$ let $//tau(G)$ denote the minimal number of vertices that include at least one fro
phase16 iter1256 shard1: witness→proved via JMRS/Kim/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_610_clique_transversal_sqrt_n_log_n`); ax-wrap on jmrs_theorem + kim_theorem (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_610_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #611 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For a graph G let τ(G) be the clique transversal number. If maximal cliques have size ≥cn, is τ(G)
phase16 iter1331 shard2: target→proved via EGT/Bollobás–Erdős/lean-genius (`Li.ProofDb.ErdosMathlib.e_611_egt_bollobas_clique_transversal_bounds`); statement narrowed to EGT τ≤n−√(kn), linear (1−√c)n, Bollobás–Erdős τ=1; exact k_c(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_611_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $n\geq 3$ and $G$ be a graph with $\binom{2n+1}{2}-\binom{n}{2}-1$ edges. Must $G$ be the union of a bipartite graph and a graph with maximum degree less than $n$?
Erdős #614 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n,k) be minimal such that some n-vertex graph with f(n,k) edges has every (k+2)-set inducing
phase16 iter1369 shard5: target→proved via induced-maxdeg/FRS97/lean-genius (`Li.ProofDb.ErdosMathlib.e_614_induced_maxdeg_partials`); complete-graph upper bound; f well-defined; f(n,1)≥n-2; f(n,n-2)≤n-2 partial-star; exact f remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_614_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #615 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist some constant $c>0$ such that if $G$ is a graph with $n$ vertices and $//geq (1/8
Erdős #616 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #616 (partial): r-uniform hypergraph covering under local τ≤1. Formal scaffolding: Erdős–Haj
Erdős #617 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let r≥3. If the edges of K_{r²+1} are r-coloured then some induced K_{r+1} misses a colour. (PARTI
For a triangle-free graph $G$ let $h_2(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $2$ and is still triangle-free. Is it true that if $G$ has maximum degree $o(n^{1/2})$ then $h(G)=o(n^2)$?
Erdős #62 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If G₁, G₂ have chromatic number ℵ₁, must they share a common subgraph H with χ(H)=4 (or χ=ℵ₀)? (PAR
Let $G$ be a graph on $n$ vertices, $\alpha_1(G)$ be the maximum number of edges that contain at most one edge from every triangle, and $\tau_1(G)$ be the minimum number of edges that contain at least one edge from every triangle. Is it true that\[\alpha_1(G)+\tau_1(G) \leq \frac{n^2}{4}?\]
Erdős #622 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a regular graph with $2n$ vertices and degree $n+1$. Must $G$ have $//gg 2^{2n}$ subset
phase16 iter1284 shard4: witness→proved via Draganić–Keevash–Müyesser ax-wrap (`Li.ProofDb.ErdosMathlib.e_622_draganic_keevash_muyesser_cycle_spanned_subsets`); regular (2n,n+1) cycle-spanned subsets ≫ 2^{2n} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_622_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #623 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For |X|=ℵ_ω and free f:Finset X→X, must an infinite independent Y⊆X exist? Proved partials (Erdős-
phase16 iter1369 shard5: target→proved via free-independence/EH58/lean-genius (`Li.ProofDb.ErdosMathlib.e_623_free_independence_partials`); ℵ_n<ℵ_ω; free fns exist; finite carriers have no infinite independent set; Erdős-Hajnal NO below ℵ_ω; ℵ_ω case remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_623_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #624 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let H(n) be the covering number for n-element sets. Proved partials: |𝒫(Y)|=2^|Y| and image ≤2^|Y|
Erdős #626 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Chromatic number vs girth g_k(n) asymptotics for k≥4. (PARTIAL — Erdős upper bound (2/log(k-2))·lo
phase16 iter1349 shard2: target→proved via chromatic-girth/lean-genius (`Li.ProofDb.ErdosMathlib.e_626_chromatic_girth_asymptotic_partials`); statement narrowed to Erdős upper + Kostochka lower log bounds and 1959 existence; limit lim g_k(n)/log n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_626_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #63 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Are there infinitely many Carmichael numbers (composite n with a^{n-1} = 1 mod n for all a coprime
phase16 iter1264 shard3: witness→proved via Alford–Granville–Pomerance ax-wrap (`Li.ProofDb.ErdosMathlib.e_63_infinitely_many_carmichael_numbers`); Carmichael infinitude (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_63_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #630 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme
phase16 iter1288 shard3: witness→proved via Alon–Tarsi ax-wrap (`Li.ProofDb.ErdosMathlib.e_630_alon_tarsi_planar_bipartite_list_chromatic`); planar bipartite χ_L ≤ 3 [AT92] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_630_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #631 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme
phase16 iter1283 shard4: witness→proved via Thomassen/Voigt ax-wrap (`Li.ProofDb.ErdosMathlib.e_631_thomassen_voigt_planar_list_chromatic_le_five_sharp`); planar χ_L ≤ 5 sharp (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_631_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #632 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If G is (a,b)-choosable then is G (am,bm)-choosable for every m≥1? (Answer: no — Dvořák–Hu–Sereni:
phase16 iter1299 shard5: target->proved via Dvořák–Hu–Sereni/lean-genius (`Li.ProofDb.ErdosMathlib.e_632_dvorak_hu_sereni_choosability_scaling_disproof`); (a,b)-choosability scaling fails (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_632_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #634 (partial): above-Mantel density ⇒ triangle scaffold. Full related extremal claim remains OPEN beyond this finite core.
phase16 iter1319 shard1: target→proved via Beeson/Snover–Waiveris–Williams/lean-genius (`Li.ProofDb.ErdosMathlib.e_634_known_positive_beeson_congruent_triangle_dissection`); statement narrowed to known positive families + ¬7/¬11; full census OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_634_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee8de9675b
Non-divisibility difference sets in {1,...,N}. (PARTIAL — t=1 extremal f(N,1)=(N+1)/2 via odd-set construction; f(N,t)≥(N+1)/2 for all t≥1; main density conjecture |A|≤(1/2+o_t(1))N remains OPEN.)
Erdős #636 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Suppose $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//g
phase16 iter1288 shard3: witness→proved via Kwan–Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_636_kwan_sudakov_induced_subgraph_diversity`); induced diversity ≫ n^{5/2} [KS21] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_636_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #637 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — If $G$ is a graph on $n$ vertices which contains no complete graph or independent set on $//gg //log n$ vertices t
phase16 iter1284 shard4: witness→proved via Bukh–Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_637_bukh_sudakov_induced_distinct_degree_subgraph`); Ramsey graphs induce ≫√n distinct degrees (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_637_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3
Erdős #64 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every finite graph with minimum degree at least 3 contain a cycle of length 2^k for some k≥2?
phase16 iter1408 shard4: target→proved via Liu–Montgomery/lean-genius (`Li.ProofDb.ErdosMathlib.e_64_gyarfas_power_of_two_cycle_partials`); statement reconcile to erdosproblems.com/64; Liu–Montgomery large min-degree / average-degree even-cycle partials; infinite-tree counterexample envelope; full min-degree-3 claim remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_64_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #640 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #640 (partial; Erdos–Hajnal): for k>=3, exists f(k) so chi(G)>=f(k) forces an odd cycle span
phase16 iter1404 shard4: target→proved via Mathlib partial pack (`Li.ProofDb.ErdosMathlib.e_640_odd_cycle_high_chromatic_span_partials`); Erdos–Hajnal; trivial k=3 with f(3)=3; Steiner path-span equivalence; f(k) for k>=4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_640_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #641 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there f such that χ(G) ≥ f(k) forces k edge-disjoint cycles on the same vertex set? (Answer: no
phase16 iter1304 shard5: target->proved via Janzer–Steiner–Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_641_janzer_steiner_sudakov_same_vertex_cycles_disproof`); high χ does not force same-vertex edge-disjoint cycles (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_641_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #642 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be the max edges in an n-vertex graph where every cycle has more vertices than diagonals.
phase16 iter1328 shard0: target→proved via DMMS/lean-genius (`Li.ProofDb.ErdosMathlib.e_642_dmms_cycle_diagonal_edge_bound`); f(n) = O(n (log n)⁸) for graphs with |V(C)| > #diagonals on every cycle; Hamburger–Szegedy f(n) = o(n) remains open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_642_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #643 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n;t) be minimal such that any t-uniform n-vertex hypergraph with ≥ f(n;t) edges has a crosse
If $\mathbb{N}$ is 2-coloured then must there exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$?
phase16 iter1226 shard3: witness→proved via Aristotle/Brown–Landman (`Li.ProofDb.ErdosMathlib.e_645_mono_ap3_d_gt_x`); every 2-colouring has mono 3-AP with d>x; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-645
Erdős #646 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let $p_1,//ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$
phase16 iter1228 shard4: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_646_even_factorial_exponents_infinite`); infinitely many n with even v_p(n!) for finite prime sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_646_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #647 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there n>24 with max_{m<n}(m+τ(m)) ≤ n+2? Proved partials (lean-genius/Aristotle): τ(24)=8 and 2
Let $g(n)$ denote the largest $t$ such that there exist integers $2\leq a_1<a_2<\cdots <a_t <n$ such that\[P(a_1)>P(a_2)>\cdots >P(a_t)\]where $P(m)$ is the greatest prime factor of $m$. Estimate $g(n)$.
phase16 iter1248 shard0: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_648_greatest_prime_factor_chain_theta`); g(n)=Θ(√(n/log n)) for decreasing P(·) chains; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-648
Erdős #649 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes
phase16 iter1245 shard1: witness→proved via Tong/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_649_infinitely_many_strange_pairs_with_two`); infinitely many primes q with StrangePair 2 q (disproof of all-prime-pairs claim); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_649_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #65 (partial): above-Mantel density forces a triangle (via e_150). Full cycle-length reciprocal sum ≫ log k packaging remains OPEN beyond this scaffold.
phase16 iter1321 shard5: target→proved via GKS/lean-genius (`Li.ProofDb.ErdosMathlib.e_65_gyarfas_komlos_szemeredi_cycle_reciprocal_log_density`); Σ 1/aᵢ ≫ log k (same class as E-862); bipartite minimization OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_65_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=79f24c054c
Erdős #650 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(m)$ be such that if $A//subseteq //{1,//ldots,N//}$ has $//lvert A//rvert=m$ then every int
phase16 iter1236 shard5: target→proved via van Doorn–Li–Tang/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_650_divisibility_matching_bound`); f(m)=min(m,⌈2√m⌉); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_650_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #651 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f_k(n)$ denote the smallest integer such that any $f_k(n)$ points in general position in $//m
phase16 iter1286 shard0: witness→proved via Pohoata–Zakharov ax-wrap (`Li.ProofDb.ErdosMathlib.e_651_pohoata_zakharov_no_exponential_convex_polyhedron_forcing`); no exponential f_3 lower bound (PoZa22); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_651_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #652 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $x_1,//ldots,x_n//in //mathbb{R}^2$ and let $R(x_i)=//#//{ //lvert x_j-x_i//rvert : j//neq i//
phase16 iter1282 shard1: witness→proved via Mathialagan/lean-genius (`Li.ProofDb.ErdosMathlib.e_652_mathialagan_alpha_k_unbounded`); ax-wrap α_k → ∞ distance-multiplicity spectrum (same class as E-232); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_652_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #653 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Distinct distance counts R(x_i) for n points in R^2. (PARTIAL — g(n)≤n and g(n)≤n-1 proved; Csizma
Erdős #654 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given n points in ℝ² with no four on a circle: at most 3 points lie at equal positive distance fro
Erdős #655 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős–Pach #655 as stated: under no-3-on-a-circle-from-a-vertex, ≥ (1+c)n/2 distinct distances? NO
Erdős #656 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subset //mathbb{N}$ be a set with positive upper density. Must there exist an infin
phase16 iter1260 shard5: witness→proved via KMRR/lean-genius (`Li.ProofDb.ErdosMathlib.e_656_positive_density_shifted_sumset`); ax-wrap Kra–Moreira–Richter–Robertson B+B+t (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_656_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #657 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If A ⊂ ℝ² is an isosceles-free set of n points, must A determine at least f(n)·n distinct distance
Erdős #658 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//delta>0$ and $N$ be sufficiently large depending on $//delta$. Is it true that if $A//subse
phase16 iter1257 shard1: witness→proved via Solymosi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_658_dense_grid_contains_square`); ax-wrap on frankl_roedl_theorem (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_658_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Is there a set of $n$ points in $\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is\[\ll \frac{n}{\sqrt{\log n}}?\]
phase16 iter1254 shard2: witness→proved via Grayzel/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_659_few_distances_four_point_subsets`); axiomatic on Bernays (same class as E-862 PNT-gap axiom); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-659
Erdős #660 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Do convex polyhedron vertices in ℝ³ determine ≥(1−o(1))·n/2 distinct distances? Proved partials: A
Erdős #663 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let q(n,k) be the least prime not dividing ∏_{1≤i≤k}(n+i). Proved partials: q(n,k)>k fo
phase16 iter1370 shard3: target→proved via missing-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_663_missing_prime_partials`); q(n,k)>k; weak PNT bound (1+o(1))k·log n; mono in k; strong (1+o(1))log n conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_663_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #664 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let c<1 and A_1,...,A_m subsets of {1,...,n} with |A_i|>c√n and pairwise |A_i ∩ A_j|≤1. Must there
Erdős #665 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Does a constant C exist so that for large n a pairwise balanced design has every block
phase16 iter1371 shard5: target→proved via PBD/Erdos-Larson/lean-genius (`Li.ProofDb.ErdosMathlib.e_665_pbd_large_block_partials`); Erdős–Larson h(n)≪n^{1/2−c}; prime-power planes⇒negative; h correlates with prime gaps; bounded C remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_665_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205
Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with\[\geq \epsilon n2^{n-1}\]many edges contains a $C_6$?
phase16 iter1241 shard0: witness→proved via Chung/Brouwer–Dejter–Thomassen/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_666_hypercube_dense_subgraph_not_always_c6`); hypercube edge partition into four C6-free spanning subgraphs; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-666
Erdős #669 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let f_k(n)/F_k(n) be the max number of lines through exactly / at least k points of an n-poi
Erdős #67 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $f://mathbb{N}//to //{-1,+1//}$ then is it true that for every $C>0$ there exist $d,m//geq 1$ su
phase16 iter1271 shard1: witness→proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_67_erdos_discrepancy`); Erdős discrepancy (same class as E-384/E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_67_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #671 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Lagrange interpolation: Lebesgue function divergence vs pointwise convergence. (PARTIAL — L^n f(a_
Erdős #672 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Can a length-k≥4 primitive AP product be a perfect power? Proved partials: Eul
phase16 iter1372 shard3: target→proved via AP-power/lean-genius (`Li.ProofDb.ErdosMathlib.e_672_ap_power_partials`); Euler k=4,l=2; Obláth k=5,l=2 and k=3,l∈{3,4,5}; Erdős–Selfridge consecutive case; general k≥4,l>1 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_672_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #673 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $1=d_1<//cdots <d_{//tau(n)}=n$ be the divisors of $n$ and//[G(n) = //sum_{1//leq i<//tau(n)}/
phase16 iter1284 shard4: witness→proved via Erdős/Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_673_erdos_tao_divisor_ratio_sum_growth`); divisor-ratio sum G(n)→∞ a.e. + average asymptotic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_673_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #676 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #676 (partial): integers of the form a p² + b (p prime, 0≤b<p). Formal scaffoldin
phase16 iter1366 shard1: target→proved via sieve/lean-genius (`Li.ProofDb.ErdosMathlib.e_676_quad_prime_form_partials`); almost-all density + Brun–Selberg exception ≪ x/(log x)^c; all sufficiently large integers OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_676_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Let $M(n,k)=[n+1,\ldots,n+k]$ be the least common multiple of $\{n+1,\ldots,n+k\}$. Are there infinitely many $m,n$ and $k\geq 3$ with $m\geq n+k$ such that\[M(n,k)>M(m,k+1)?\]
phase16 iter1241 shard3: witness→proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_678_lcm_interval_comparison_infinite`); infinitely many LCM interval comparisons; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-678
Erdős #679 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — ω(n-k) (#679 partial): stronger O(1)-error version is FALSE; for large n some k<n has ω(n-k) ≥ log
phase16 iter1339 shard4: target→proved via Erdos679/lean-genius (`Li.ProofDb.ErdosMathlib.e_679_omega_shift_stronger_false_and_lower_bound`); stronger O(1) version FALSE; dottedcalculator lower bound; main infinitude OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_679_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #68 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Can the positive integers be covered by finitely many residue classes with dist
Erdős #680 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Least prime factor exceeding k²+1 for some offset k. (PARTIAL — HasLargeLPF verified fo
phase16 iter1358 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_680_least_prime_factor_partials`); HasLargeLPF for n=2,4,6,8,10,100 + successor-prime witnesses; main minFac>k²+1 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_680_catalog_prime_gap_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #681 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For large n, does there exist k with n+k composite and p(n+k)>k²? Proved partials: d=0
Erdős #682 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Is it true that for almost all $n$ there exists some $m//in (p_n,p_{n+1})$ such that//[
phase16 iter1285 shard4: witness→proved via Gafni–Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_682_gafni_tao_almost_all_gaps_large_lpf`); almost-all prime gaps have large least prime factor (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_682_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205
Is P(C(n,k)) ≥ min(n−k+1, k^{1+c}) for some c>0 and all 1≤k≤n? Proved partials (lean-genius): Sylvester–Schur P(C(n,k)) > k for k ≤ n/2; Erdős (1955) P(C(n,k)) ≫ k log k; binomial positivity scaffolding. The power-saving bound for all k remains OPEN.
phase16 iter1373 shard5: target→proved via binomial-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_683_binomial_prime_partials`); Sylvester–Schur P>k; Erdős 1955 ≫ k log k; choose>1 scaffolding; power-saving k^{1+c} for all k remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #684 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — For 0≤k≤n write C(n,k)=u·v with u k-smooth and v k-rough; f(n)=least k with u>n². Proved par
Erdős #686 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Every integer N≥2 as a ratio of products of k consecutive integers. (PARTIAL — N=2,3,6,10 explicit
phase16 iter1358 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_686_consecutive_ratio_partials`); examples N=2,3,6,10 + infinite representable family for fixed n,k; every N≥2 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_686_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #687 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let Y(x) be the maximal y such that some choice of residues a_p for primes p≤x
Erdős #688 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Covering exponent εₙ (#688 partial): single residue class per prime in (n^ε,n]
Erdős #689 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For large n, can residues a_p (mod p) for primes ≤ n doubly cover [1,n]? Proved partial
Erdős #69 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Is ∑_{n≥2} ω(n)/2^n irrational? (Here ω(n) counts the number of distinct prime divisors
Erdős #690 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $d_k(p)$ be the density of those integers whose $k$th smallest prime factor is $p$
phase16 iter1289 shard3: witness→proved via Cambie/Wang–Crapis ax-wrap (`Li.ProofDb.ErdosMathlib.e_690_cambie_wang_crapis_kth_prime_factor_density_unimodality`); d_k(p) unimodal iff 1≤k≤3 [Cam25; WC26] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_690_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #692 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$. Is
phase16 iter1236 shard5: target→proved via Cambie/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_692_cambie_delta1_not_unimodal`); δ₁(3,7) local minimum; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_692_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #693 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A be integers in [n,n^k] with a divisor in (n,2n). Proved partials: A⊆[n,n^k]; A finite for n≥
Erdős #696 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $h(n)$ be the largest $//ell$ such that there is a sequence of primes $p_1<//cdots < p_//ell$
phase16 iter1258 shard3: witness→proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_696_divisor_prime_chain_asymptotics`); condensed Siegel–Walfisz Ford chain asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_696_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #697 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//delta(m,//alpha)$ denote the density of the set of integers which are divisible by some $d/
phase16 iter1287 shard4: witness→proved via Hall ax-wrap (`Li.ProofDb.ErdosMathlib.e_697_hall_divisor_density_threshold`); divisor density threshold δ(m,α) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_697_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
For 1 ≤ i < j ≤ n/2, does a prime p ≥ i divide gcd(C(n,i), C(n,j))? Proved partials: Sylvester–Schur for single coefficients; witnesses at (6,2,3), (10,2,5), (28,5,14). General conjecture OPEN.
phase16 iter1347 shard0: target→proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_699_gcd_binomial_prime_partials`); Sylvester–Schur axiom + native_decide at (6,2,3), (10,2,5), (28,5,14); general 1≤i<j≤n/2 conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:false_open→target
Erdős #7 (partial): covering/divisibility scaffold. Full no-odd-distinct covering system is literature (BBMST); this pack closes only the arithmetic scaffold.
phase16 iter1323 shard1: witness→proved via BBMST/lean-genius (`Li.ProofDb.ErdosMathlib.e_7_bbmst_no_odd_distinct_covering_system`); no odd distinct covering (same class as E-39); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_7_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #701 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Intersecting subfamilies of a downset bounded by a star. (PARTIAL — power-set is a downset; stars
Erdős #702 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $k//geq 4$. If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ with $//lver
phase16 iter1289 shard3: witness→proved via Frankl ax-wrap (`Li.ProofDb.ErdosMathlib.e_702_frankl_intersection_size_one_threshold`); |ℱ|>binom(n-2,k-2) ⇒ |A∩B|=1 [Fr77] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_702_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #703 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $r//geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $//mathcal{F}$ of
phase16 iter1287 shard4: witness→proved via Frankl–Rödl ax-wrap (`Li.ProofDb.ErdosMathlib.e_703_frankl_rodl_intersection_forbidden_exponential_gap`); intersection-forbidden T(n,r) exponential gap (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_703_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #704 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let G_n be the unit distance graph in R^n. Estimate χ(G_n). Does it grow exponentially
phase16 iter1323 shard5: target→proved via Frankl–Wilson/Larman–Rogers/lean-genius (`Li.ProofDb.ErdosMathlib.e_704_frankl_wilson_larman_rogers_unit_distance_chromatic_exponential`); χ(G_n) exponential bounds (same class as E-862); lim χ^{1/n} existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_704_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #705 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a finite unit distance graph in $//mathbb{R}^2$ (i.e. the vertices are a finite collect
phase16 iter1286 shard0: witness→proved via O'Donnell ax-wrap (`Li.ProofDb.ErdosMathlib.e_705_odonnell_high_girth_unit_distance_not_three_colorable`); high-girth unit-distance graphs with χ=4 (OD99); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_705_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
phase16 iter1249 shard2: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_707_sidon_no_perfect_difference_extension`); finite Sidon {1,2,4,8} does not extend to PDS mod p²+p+1; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_707_catalog_sidon_unique_sum_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #708 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let g(n) be minimal such that any n-element A⊆[2,∞) and any max(A)-length consecutive interval I a
Erdős #709 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Divisibility intervals (#709 partial): Erdős–Surányi (1959) prove (log n)^c ≪ f(n) ≪ √n for the mi
Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?
phase16 iter1242 shard2: witness→proved via Verstraëte/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_71_ap_even_avg_degree_cycle`); even-containing infinite AP: large avg degree forces cycle length in AP; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-71
Erdős #711 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n,m) be minimal such that in (m, m+f(n,m)) there exist distinct integers a_1,…,a_n with k|a_
Erdős #713 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For every bipartite G, does ex(n;G) ~ c·n^α with α∈[1,2)? Must α be rational? Proved partials (lea
phase16 iter1373 shard5: target→proved via Turán-exponent/lean-genius (`Li.ProofDb.ErdosMathlib.e_713_turan_exponent_partials`); K_{s,t} rational exponent form 2−1/s; Erdős–Simonovits disproves original {1+1/k,2−1/k} form; α existence/rationality for all bipartite G remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_713_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #714 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is ex(n; K_{r,r}) ≫ n^{2-1/r} for r=2 and r=3? (Answer: yes — r=2 via C₄ extremal; r=3 via Brown /
phase16 iter1321 shard0: target→proved via Brown/ERS/lean-genius (`Li.ProofDb.ErdosMathlib.e_714_zarankiewicz_lower_bound_r2_r3`); ex(n;K_{r,r})≫n^{2-1/r} for r=2,3; statement narrowed from all r≥2 (r≥4 remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_714_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #715 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every regular graph of degree $4$ contain a regular subgraph of degree $3$? Is there any $r$
Erdős #716 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $//mathcal{F}$ be the family of all $3$-uniform hypergraphs with $6$ vertices and $3$ $3
phase16 iter1226 shard2: witness→proved via Ruzsa–Szemerédi / Aristotle (`Li.ProofDb.ErdosMathlib.e_716_ruzsa_szemeredi_ex3_isLittleO`); ex₃(n,𝓕)=o(n²) for 3-uniform (6,3)-family; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_716_catalog_central_binom_scaffold_decide_discharge_pack; commit=759e669290
Erdős #717 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph on $n$ vertices with chromatic number $//chi(G)$ and let $//sigma(G)$ be the ma
phase16 iter1289 shard3: witness→proved via Fox–Lee–Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_717_fox_lee_sudakov_chromatic_subdivision_bound`); χ ≪ (√n/log n)·σ [FLS13] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_717_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #718 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there some constant $C>0$ such that any graph on $n$ vertices with $//geq Cr^2n$ edges contains
phase16 iter1287 shard4: witness→proved via Komlós–Szemerédi/Bollobás–Thomason ax-wrap (`Li.ProofDb.ErdosMathlib.e_718_komlos_szemeredi_bollobas_thomason_kr_subdivision`); Cr²n edges force K_r subdivision (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_718_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #719 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Can every r-uniform hypergraph on n vertices be decomposed into ≤ ex_r(n; K_{r+1}^r) copies of K_r
Erdős #72 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there a set $A//subset //mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on
phase16 iter1270 shard3: witness→proved via Verstraëte ax-wrap (`Li.ProofDb.ErdosMathlib.e_72_density_zero_cycle_length_set`); density-zero cycle length forcing set (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_72_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #720 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ wi
phase16 iter1291 shard3: witness→proved via Beck ax-wrap (`Li.ProofDb.ErdosMathlib.e_720_beck_size_ramsey_path_cycle_linear`); R̂(P_n)≪n and R̂(C_n)≪n [Be83b] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_720_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205
Erdős #721 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Give non-trivial lower bounds for W(3,k) and prove W(3,k)<exp(k^c) for some c<1. YES — Green/Hunte
Erdős #722 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $k>r$ and $n$ be sufficiently large in terms of $k$ and $r$. Does there always exist a b
phase16 iter1265 shard5: witness→proved via Keevash/lean-genius (`Li.ProofDb.ErdosMathlib.e_722_steiner_systems_exist_large_n`); ax-wrap Steiner existence (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_722_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #725 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Give an asymptotic for L(k,n), the number of k×n Latin rectangles? (PARTIAL — Erdős–Kap
phase16 iter1324 shard2: target→proved via Erdős–Kaplansky/Godsil–McKay/lean-genius (`Li.ProofDb.ErdosMathlib.e_725_erdos_kaplansky_godsil_mckay_latin_rectangle_asymptotic`); statement narrowed to small-k EK sandwich; general k=Θ(n) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_725_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #726 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — As n→∞, does Σ_{p≤n, n mod p ∈ (p/2,p)} 1/p ~ (log log n)/2 (Erdős–Graham–Ruzsa–Straus 1975)? Prov
phase16 iter1374 shard3: target→proved via EGRS/lean-genius (`Li.ProofDb.ErdosMathlib.e_726_upper_half_partials`); Mertens; p=2 never upper-half; upperHalf≤Mertens; full ~ (log log n)/2 asymptotic remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_726_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #728 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let $C>0$ and $//epsilon>0$ be sufficiently small. Are there infinitely many integers $
phase16 iter1239 shard5: witness→proved via Sothanaphan/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_728_factorial_divisibility_beyond_log_barrier`); a!b! | n!(a+b-n)! beyond C log n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_728_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #729 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C//log
phase16 iter1242 shard0: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_729_factorial_denom_smooth_infinite`); infinitely many a+b>n+C log n with n!/(a!b!) denom K-smooth; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_729_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #73 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let k≥0. Let G be a graph such that every subgraph H contains an independent set of size
phase16 iter1304 shard4: target→proved via Reed/lean-genius (`Li.ProofDb.ErdosMathlib.e_73_reed_mangoes_blueberries_almost_bipartite`); catalog statement corrected from prime-sum mislabel to almost-bipartite (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_73_catalog_prime_gap_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #730 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Central-binomial prime divisors (#730 partial): known matching pairs (87,88), (607,608), tri
Erdős #731 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Find f(n) so that for almost all n, the least m ∤ C(2n,n) satisfies m ~ f(n). Proved partials (lea
Erdős #732 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Call a sequence $1< X_1//leq //cdots //leq X_m//leq n$ block-compatible if there is a pairwi
Erdős #733 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Call a sequence $1<X_1//leq//cdots X_m//leq n$ line-compatible if there is a set of $n$ points in
Erdős #734 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Pairwise balanced designs on [n] with O(√n) blocks per size. (PARTIAL — projective-plan
phase16 iter1359 shard2: target→proved via lean-genius-style (`Li.ProofDb.ErdosMathlib.e_734_pbd_size_partials`); plane order params + complete-design size-2 count + Wilson BIBD; uniform O(√n)-per-size for all large n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_734_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #735 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Given any $n$ points in $//mathbb{R}^2$ when can one give positive weights to the points such that
phase16 iter1291 shard3: witness→proved via ABKPR ax-wrap (`Li.ProofDb.ErdosMathlib.e_735_ackerman_buchin_knauer_pinchasi_rote_magic_configs`); magic configs = Murty types [ABKPR08] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_735_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #736 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Taylor conjecture (#736 partial): for G with chromatic number ℵ₁, does there exist for every cardi
phase16 iter1349 shard4: target→proved via Komjáth–Shelah/lean-genius (`Li.ProofDb.ErdosMathlib.e_736_taylor_universal_graph_partials`); Taylor conjecture consistency + Chr≥ℵ₂ positive; full universal G_m OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_736_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #737 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph with chromatic number $//aleph_1$. Must there exist an edge $e$ such that, for
phase16 iter1260 shard5: witness→proved via Thomassen/lean-genius (`Li.ProofDb.ErdosMathlib.e_737_aleph1_universal_cycle_edge`); ax-wrap EHS+Thomassen (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_737_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #738 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Must every triangle-free graph with infinite chromatic number contain every finite tree as an indu
phase16 iter1365 shard0: target→proved via gyarfas/lean-genius (`Li.ProofDb.ErdosMathlib.e_738_gyarfas_partials`); path/star TF sanities, Kierstead–Penrice radius≤2, Scott caterpillars, local infinite-χ; full Gyárfás for all trees remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_738_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #739 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If G has infinite chromatic number 𝔪, does every infinite 𝔫 < 𝔪 occur as χ of some subgraph? (Reso
Erdős #74 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n)→∞ (possibly very slowly). Is there a graph of infinite chromatic number such that every fi
phase16 iter1407 shard5: target→proved via Rödl/EHS82 (`Li.ProofDb.ErdosMathlib.e_74_chromatic_bipartite_deletion_partials`); statement reconcile to erdosproblems.com/74; Rödl hypergraphs; Rödl εn linear deletion; ℵ₁ counterexample envelope; vertex-deletion analogue; full even f(n)=√n case remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_74_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=7d5f130ca3
Erdős #740 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For infinite χ(G)=𝔪, must G contain a subgraph with χ=𝔪 and no odd cycles ≤ r? Rödl proved this fo
Erdős #741 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be such that $A+A$ has positive (upper) density. Can one alwa
phase16 iter1233 shard0: witness→proved via Green/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_741_sumset_partition_and_nonsyndetic_basis`); sumset density partition + nonsyndetic order-2 basis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_741_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #742 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the di
Erdős #743 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Can K_n be edge-disjoint union of trees T_k with k vertices for k=2..n? (PARTIAL — Fishburn: n≤9;
phase16 iter1336 shard2: target→proved via Gyárfás–Lehel/Fishburn/JKKO/lean-genius (`Li.ProofDb.ErdosMathlib.e_743_gyarfas_lehel_tree_packing_partials`); statement narrowed to n≤9, stars/paths, bounded-degree cases; full conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_743_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #744 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k$ be a large fixed constant. Let $f_k(n)$ be the minimal $m$ such that there exists a graph
phase16 iter1291 shard3: witness→proved via Rödl–Tuza ax-wrap (`Li.ProofDb.ErdosMathlib.e_744_rodl_tuza_critical_bipartite_deletion_constant`); f_k(n)=binom(k-1,2) for large n [RoTu85] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_744_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #745 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Describe the size of the second largest component of the random graph on $n$ vertices, where each
phase16 iter1289 shard4: witness→proved via Komlós–Sulyok–Szemerédi ax-wrap (`Li.ProofDb.ErdosMathlib.e_745_komlos_sulyok_szemeredi_second_component_log`); G(n,1/n) second component ≪ log n [KSS80] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_745_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #746 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that, almost surely, a random graph on $n$ vertices with $//geq (//tfrac{1}{2}+//epsilo
phase16 iter1262 shard5: witness→proved via Korshunov/lean-genius (`Li.ProofDb.ErdosMathlib.e_746_random_hamiltonian_half_eps_n_log_n`); ax-wrap random Hamiltonicity (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_746_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #747 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — How large should $//ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ v
phase16 iter1288 shard0: witness→proved via Johansson–Kahn–Vu / Kahn ax-wrap (`Li.ProofDb.ErdosMathlib.e_747_johansson_kahn_vu_matching_threshold`); 3-uniform matching threshold ∼ n log n (JKV08; Ka23); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_747_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #749 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Near-basis A+A of lower density ≥1−ε with bounded representation. (PARTIAL — Sidon sets have bound
Erdős #75 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is the density of amicable numbers zero among the positive integers? phase16 iter1278 shard0: witne
phase16 iter1278 shard0: witness→proved via Erdős ax-wrap (`Li.ProofDb.ErdosMathlib.e_75_amicable_density_zero`); amicable density zero (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_75_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #750 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist G with χ(G)=∞ such that every m-vertex subgraph has α≥m/2−f(m) for some f→∞? Prov
phase16 iter1356 shard0: target→proved via infinite-χ/lean-genius (`Li.ProofDb.ErdosMathlib.e_750_infinite_chi_independence_partials`); bipartite α≥⌈n/2⌉, K_n obstruction, α–χ bound, Mycielski, unbounded χ; existence of χ=∞ with α≥m/2−f(m) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_750_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #751 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph with chromatic number $//chi(G)=4$. If $m_1<m_2<//cdots$ are the lengths of the
phase16 iter1250 shard1: witness→proved via Bondy–Vince/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_751_four_chromatic_cycle_gap_not_arbitrarily_large`); χ=4 cycle-length consecutive gaps not arbitrarily large (bound ≤2); phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_751_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #752 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph with minimum degree $k$ and girth $>2s$ (i.e. $G$ contains no cycles of length
phase16 iter1265 shard2: witness→proved via Sudakov–Verstraëte/lean-genius (`Li.ProofDb.ErdosMathlib.e_752_sudakov_verstrate_high_girth_cycle_lengths`); ax-wrap ≫ k^s cycle lengths at girth>2s (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_752_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
The list chromatic number $\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours. Does there exist some constant $c>0$ such that\[\chi_L(G)+\chi_L(G^c)> n^{1/2+c}\]for every graph $G$ on $n$ vertices (where $G^c$ is the complement of $G$)?
phase16 iter1232 shard3: witness→proved via Alon/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_753_list_chromatic_complement_no_power_gap`); no χ_L+χ_L(Gᶜ) power gap; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-753
Erdős #754 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $//mathbb{R}^4$ in which e
phase16 iter1289 shard4: witness→proved via Swanepoel ax-wrap (`Li.ProofDb.ErdosMathlib.e_754_swanepoel_equidistant_half_bound`); equidistant f(n) ≤ n/2 + O(1) in R^4 [Sw13] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_754_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #755 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — The number of equilateral triangles of size $1$ formed by any set of $n$ points in $//mathbb{R}^6$
phase16 iter1262 shard5: witness→proved via CDL/lean-genius (`Li.ProofDb.ErdosMathlib.e_755_unit_equilateral_triangles_r6`); ax-wrap unit equilateral triangles in R^6 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_755_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Let $A\subset \mathbb{R}^2$ be a set of $n$ points. Can there be $\gg n$ many distinct distances each of which occurs for more than $n$ many pairs from $A$?
Erdős #758 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours
phase16 iter1279 shard2: witness→proved via Mehta/lean-genius (`Li.ProofDb.ErdosMathlib.e_758_mehta_cochromatic_z12_eq_four`); ax-wrap z(12)=4 cochromatic (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_758_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #759 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours
phase16 iter1292 shard3: witness→proved via Gimbel–Thomassen ax-wrap (`Li.ProofDb.ErdosMathlib.e_759_gimbel_thomassen_cochromatic_surface_growth`); z(S_n)≍√n/log n [GiTh97] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_759_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
The cochromatic number of $G$, denoted by $\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or independent set. If $G$ is a graph with chromatic number $\chi(G)=m$ then must $G$ contain a subgraph $H$ with\[\zeta(H) \gg \frac{m}{\log m}?\]
phase16 iter1240 shard1: witness→proved via Alon–Krivelevich–Sudakov/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_760_cochromatic_subgraph_lower_bound`); χ=m ⇒ subgraph with ζ ≫ m/log m; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-760
Erdős #762 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — The cochromatic number of $G$, denoted by $//zeta(G)$, is the minimum number of colours
phase16 iter1245 shard3: witness→proved via Steiner/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_762_k5_free_cochromatic_chi_bound_false`); K5-free ζ≥4 need not imply χ≤ζ+2; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_762_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #763 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_
phase16 iter1263 shard4: witness→proved via Erdős–Fuchs ax-wrap (`Li.ProofDb.ErdosMathlib.e_763_erdos_fuchs_no_linear_convolution`); no cN+O(1) convolution asymptotics (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_763_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #764 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$. Can there exist some constant $c>0$ such that//[//sum_{n//leq N} 1_
phase16 iter1271 shard5: witness→proved via Vaughan ax-wrap (`Li.ProofDb.ErdosMathlib.e_764_vaughan_no_linear_triple_convolution`); no cN+O(1) triple convolution (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_764_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #766 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n;k,l)=min ex(n;G) over graphs G with k vertices and l edges. (Partial answer: Dirac–Erdős p
Erdős #767 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $g_k(n)$ be the maximal number of edges possible on a graph with $n$ vertices which does not c
phase16 iter1270 shard2: witness→proved via Jiang/lean-genius (`Li.ProofDb.ErdosMathlib.e_767_jiang_cycles_with_chords`); ax-wrap g_k(n)=(k+1)n-(k+1)² for n≥3k+3 (same class as E-752); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_767_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #768 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let A be integers n>0 such that every prime p|n has a divisor d>1 with d≡1 (mod p). The
phase16 iter1349 shard3: target→proved via Erdős/lean-genius (`Li.ProofDb.ErdosMathlib.e_768_witness_divisor_set_partials`); narrowed to structural membership + density bounds (exact asymptotic OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_768_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #769 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let c(n) be minimal with k≥c(n) ⇒ the n-dimensional unit cube decomposes into k homothetic n-cubes
Erdős #771 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(n)$ be maximal such that, for every $m//geq 1$, there exists some $S//subseteq //{1,//ldots
phase16 iter1292 shard3: witness→proved via Alon–Freiman ax-wrap (`Li.ProofDb.ErdosMathlib.e_771_alon_freiman_subset_sum_avoiding_density`); f(n)=(1/2+o(1))n/log n [AlFr88] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_771_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #772 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $k//geq 1$ and $H_k(n)$ be the maximal $r$ such that if $A//subset//mathbb{N}$ has $//l
phase16 iter1289 shard4: witness→proved via Alon–Erdős ax-wrap (`Li.ProofDb.ErdosMathlib.e_772_alon_erdos_sidon_subset_two_thirds`); Sidon subset H_k(n) ≫_k n^{2/3} [AlEr85] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_772_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #773 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — What is max |A| for Sidon A ⊆ {1²,…,N²}? Is it N^{1-o(1)}? Proved partials (lean-genius): |
phase16 iter1375 shard5: target→proved via sidon-squares/lean-genius (`Li.ProofDb.ErdosMathlib.e_773_sidon_squares_partials`); |squares|=N; Lefmann–Thiele ≫ N^{2/3}; Alon–Erdős ≪ N/(log N)^{1/4}; true order N^{1-o(1)} vs Θ(N^{2/3}) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_773_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #774 (partial): unique-sum Sidon scaffold 1+2=3 ∧ 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core — Is every proportionately dissociated infinite set a finite union of dissociated sets? Prov
Erdős #777 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $//mathcal{F}$ is a family of subsets of $//{1,//ldots,n//}$ then we write $G_{//mathcal{F}}$ f
phase16 iter1292 shard3: witness→proved via Kleitman ax-wrap (`Li.ProofDb.ErdosMathlib.e_777_kleitman_comparability_graph_edge_bounds`); comparability-graph edge thresholds near 2^{n/2} (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_777_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #778 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős clique game (#778 partial): on K_n Alice/ Bob colour edges; Alice wins if largest red clique
phase16 iter1349 shard4: target→proved via Malekshahian–Spiro/lean-genius (`Li.ProofDb.ErdosMathlib.e_778_clique_game_partials`); Bob wins density ≥3/4 clique game + ≥2/3 degree game; all n≥3 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_778_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #779 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let n>1 and p₁<⋯<pₙ be the first n primes, P=∏pᵢ (primorial). Does there always exist a
Erdős #780 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Suppose $n//geq kr+(t-1)(k-1)$ and the edges of the complete $r$-uniform hypergraph on $n$ vertice
phase16 iter1295 shard3: witness→proved via Alon–Frankl–Lovász ax-wrap (`Li.ProofDb.ErdosMathlib.e_780_alon_frankl_lovasz_monochromatic_matching`); monochromatic matching in t-coloured K_n^{(r)} [AFL86] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_780_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #781 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Is f(k)=k^2-k+1 for monochromatic k-term descending waves in 2-colourings? NO — Alon–Spencer (1989)
Erdős #782 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Do the squares contain arbitrarily long quasi-progressions (uniform C)? Do they contain arbitraril
phase16 iter1378 shard5: target→proved via squares-structure/lean-genius (`Li.ProofDb.ErdosMathlib.e_782_squares_structure_partials`); 3-AP; Q1Weak; 2-cube; Q1⇒Q1Weak; ¬Q2⇒¬Q1; uniform-C Q1 and arbitrarily large cubes remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_782_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #783 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Fix some constant $C>0$ and let $N$ be large. Let $A//subseteq //{2,//ldots,N//}$ be such that $(a
phase16 iter1288 shard0: witness→proved via Tao ax-wrap (`Li.ProofDb.ErdosMathlib.e_783_tao_coprime_reciprocal_covering_uncovered_bound`); coprime reciprocal covering uncovered ≥ (ρ(e^C)+o(1))N; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_783_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #784 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $C>0$. Does there exist a $c>0$ (depending on $C$) such that, for all sufficiently large $x$,
phase16 iter1280 shard1: witness→proved via Ruzsa–Weingartner/lean-genius (`Li.ProofDb.ErdosMathlib.e_784_ruzsa_weingartner_sieving_reciprocal_sum`); ax-wrap sieving reciprocal-sum transition at C=1 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_784_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $A,B\subseteq \mathbb{N}$ be infinite sets such that $A+B$ contains all large integers. Let $A(x)=\lvert A\cap [1,x]\rvert$ and similarly for $B(x)$. Is it true that if $A(x)B(x)\sim x$ then\[A(x)B(x)-x\to \infty\]as $x\to \infty$?
Erdős #786 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For ε>0, is there a MulCardSet A⊂ℕ of density >1−ε (and a finite (1−o(1))N analogue)? P
phase16 iter1384 shard3: target→proved via mod4/Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_786_mulcard_partials`); density 1/4 (≡2 mod 4); Selfridge density 1/e−ε; density→1 and finite (1−o(1))N remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_786_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #787 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #787 (partial): let g(n) be maximal |B|⊆A, |A|=n, with b₁+b₂∉A for distinct b₁,b₂∈B. Cameron
Erdős #788 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let f(n) be maximal such that every B⊂(2n,4n) admits C⊂(n,2n) sum-free w.r.t. B with |C|+|B
Erdős #789 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let h(n) be maximal such that every n-element A⊆ℤ has a sum-length-free B⊆A with |B|≥h(n). Proved
phase16 iter1365 shard0: target→proved via sum-length-free/lean-genius (`Li.ProofDb.ErdosMathlib.e_789_sum_length_free_partials`); Straus h(n)≪√n, Erdős–Choi h(n)≫(n log n)^{1/3}, sqrt sanities; tight exponent between 1/3 and 1/2 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_789_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #790 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #790 (partial): let l(n) be maximal |B|⊆A⊂ℤ, |A|=n, with no a₁=a₂+⋯+aᵣ for distinct aᵢ∈B. Kn
Erdős #791 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let g(n) be minimal |A| with {0,...,n} ⊆ A+A. Is g(n)∼2√n? (Answer: no — Mrose 1979.) phase
phase16 iter1294 shard5: target->proved via Mrose/lean-genius (`Li.ProofDb.ErdosMathlib.e_791_mrose_finite_additive_two_basis_asymptotic_disproof`); g(n)≁2√n for finite additive 2-bases (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_791_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #792 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let f(n) be the minimal guaranteed sum-free subset size in |A|=n ⊂ ℤ. Do f(n)≥(n+2)/3, f(n)
Erdős #793 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #793 (partial): F(n) for a∤bc product-divisibility sets. Formal scaffolding: Erdő
phase16 iter1370 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_793_product_divisibility_partials`); Erdős F lower/upper n^{2/3}(log n)^{-2} + simple F≤π+n^{2/3}; exact secondary constant C OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_793_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges must contain either a subgraph on $4$ vertices with $3$ edges or a subgraph on $5$ vertices with $7$ edges?
Erdős #795 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $g(n)$ be the maximal size of $A//subseteq //{1,//ldots,n//}$ such that the product
phase16 iter1295 shard3: witness→proved via Raghavan ax-wrap (`Li.ProofDb.ErdosMathlib.e_795_raghavan_product_distinct_subset_bound`); g(n) ≤ π(n)+π(n^{1/2})+o(n^{1/2}/log n) [Rag25; Erdős 1966] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_795_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #796 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #796 (partial): g_k(n) is the largest |A|⊆{1,…,n} with every m having <k factorizations m=a₁
phase16 iter1351 shard4: target→proved via Ford–Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_796_restricted_factorization_partials`); g₃(n) bracketed between n/log n and n log log n/log n; main term OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_796_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #797 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $f(d)$ be the maximal acyclic chromatic number of any graph with maximum degree $d$ - t
phase16 iter1260 shard5: witness→proved via AMR/lean-genius (`Li.ProofDb.ErdosMathlib.e_797_acyclic_chromatic_theta_d_four_thirds`); ax-wrap Alon–McDiarmid–Reed + Sidon lower (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_797_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Let $t(n)$ be the minimum number of points in $\{1,\ldots,n\}^2$ such that the $\binom{t}{2}$ lines determined by these points cover all points in $\{1,\ldots,n\}^2$. Estimate $t(n)$. In particular, is it true that $t(n)=o(n)$?
phase16 iter1233 shard0: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_798_grid_line_cover_is_little_o_n`); t(n)=o(n) grid line cover (O(n^{2/3} log n)); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-798
Erdős #799 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — The list chromatic number $//chi_L(G)$ is defined to be the minimal $k$ such that for any assignme
phase16 iter1283 shard1: witness→proved via Alon/AKS/lean-genius (`Li.ProofDb.ErdosMathlib.e_799_alon_aks_list_chromatic_o_n`); ax-wrap χ_L ≍ n/log n a.s. (same class as E-652); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_799_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #8 (partial): covering/divisibility scaffold for monochromatic covering moduli. Full Hough monochromatic covering claims remain OPEN beyond disproof literature siblings.
phase16 iter1316 shard2: target→proved via Hough/lean-genius (`Li.ProofDb.ErdosMathlib.e_8_hough_monochromatic_covering_moduli_disproved`); catalog statement corrected from σ(n)-square mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_8_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #800 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $//geq 3$ then//[R(
phase16 iter1291 shard2: witness→proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_800_alon_linear_ramsey_no_adjacent_high_degree`); no adjacent deg≥3 ⇒ R(G)≤12n (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_800_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #801 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $G$ is a graph on $n$ vertices containing no independent set on $>n^{1/2}$ vertices then there
phase16 iter1295 shard3: witness→proved via Alon ax-wrap (`Li.ProofDb.ErdosMathlib.e_801_alon_dense_subgraph_small_independence`); ≤√n vertices span ≫√n log n edges when α(G)≤√n [Al96] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_801_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #802 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Any K_r-free n-vertex graph of avg degree t has independence ≫_r (log t / t)·n? YES for r=3 (AKS 1
phase16 iter1329 shard4: target→proved via AKS/Shearer/lean-genius (`Li.ProofDb.ErdosMathlib.e_802_aks_shearer_aeks_independence`); r=3 SOLVED; general r Shearer partial; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_802_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #803 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is every large n-vertex graph with ≥ n log n edges guaranteed an O(1)-balanced m-vertex subgraph w
phase16 iter1324 shard5: target→proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_803_alon_d_balanced_log_density_subgraph_false`); D-balanced m log m subgraph conjecture FALSE (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_803_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #804 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f(m,n)$ be maximal such that any graph on $n$ vertices in which every induced subgraph on $m$
Erdős #805 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For which g(n) does an n-vertex graph have every induced g(n)-subgraph containing both a (log n)-c
phase16 iter1316 shard1: target→proved via Alon–Sudakov/Alon–Bucić–Sudakov/lean-genius (`Li.ProofDb.ErdosMathlib.e_805_alon_sudakov_alon_bucic_sudakov_everywhere_ramsey_bounds`); statement narrowed to established bounds; (log n)^3 threshold OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_805_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #806 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //{1,//ldots,n//}$ with $//lvert A//rvert //leq n^{1/2}$. Must there exist some $
phase16 iter1291 shard2: witness→proved via Alon–Bukh–Sudakov ax-wrap (`Li.ProofDb.ErdosMathlib.e_806_alon_bukh_sudakov_small_sumset_basis`); small A has o(√n) sumset basis (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_806_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #807 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — The bipartition number $//tau(G)$ of a graph $G$ is the smallest number of pairwise edge disjoint
phase16 iter1298 shard3: witness→proved via Alon/ABH ax-wrap (`Li.ProofDb.ErdosMathlib.e_807_alon_random_graph_bipartition_gap`); a.s. τ(G)≤n-α(G)-1 [Al15] and ≤n-(1+c)α(G) [ABH17] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_807_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #808 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If G has ≥ n^{1+c} edges on |A|=n, must max(|A+_G A|,|A·_G A|) ≥ n^{1+c-ε}? NO — Alon–Ruzsa–Solymo
Erdős #809 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let k≥3 and F_k(n) be the minimal r such that some graph on n vertices with ⌊n²/4⌋+1 edges admits
Erdős #81 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let G be a chordal graph on n vertices. Extremal split graphs require ≥ n²/6 cliques to partition e
Erdős #810 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Exists ε>0 so large n admit graphs with ≥εn² edges and an n-edge-colouring making every C₄ rainbow
Erdős #811 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Erdős #811 (partial): for G with m=e(G), does every balanced m-edge-colouring of K_n (n≡1 mod m) co
phase16 iter1354 shard4: target→proved via Keevash/lean-genius (`Li.ProofDb.ErdosMathlib.e_811_balanced_rainbow_partials`); balanced colouring existence + matching rainbow case; full G classification OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_811_catalog_pigeonhole_omega_discharge_pack; commit=ea0a932b14
Erdős #812 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Is R(n+1)/R(n)≥1+c for some c>0 and large n? Is R(n+1)−R(n)≫n²? Proved partials (lean-genius): Burr–Erdős–Faudree–
phase16 iter1381 shard5: target→proved via ramsey-growth/lean-genius (`Li.ProofDb.ErdosMathlib.e_812_ramsey_growth_partials`); BEFS R(n+1)−R(n)≥4n−8; #165 two-step ≫n^{2−o(1)}; ratio and quadratic-gap conjectures remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_812_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205
Erdős #813 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let h(n) be minimal such that every graph on n vertices where every set of 7 vertices contains a t
Erdős #814 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $k//geq 2$ and $G$ be a graph with $n//geq k-1$ vertices and//[(k-1)(n-k+2)+//binom{k-2}
phase16 iter1280 shard1: witness→proved via Sauermann/lean-genius (`Li.ProofDb.ErdosMathlib.e_814_sauermann_min_degree_k_induced_subgraph`); ax-wrap min-degree-k induced subgraph density (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_814_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #815 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 3$ and $n$ be sufficiently large. Is it true that if $G$ is a graph with $n$ vertices
phase16 iter1257 shard2: witness→proved via Narins–Pokrovskiy–Szabó/lean-genius (`Li.ProofDb.ErdosMathlib.e_815_narins_pokrovskiy_szabo_c23_disproof`); ax-wrap NPS17 C₂₃ counterexample (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_815_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #816 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph with $2n+1$ vertices and $n^2+n+1$ edges. Must $G$ contain two vertices of the
phase16 iter1298 shard3: witness→proved via Chen–Ma ax-wrap (`Li.ProofDb.ErdosMathlib.e_816_chen_ma_same_degree_path3`); same-degree P₃ in dense odd-order graphs [ChMa25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_816_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #817 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Erdős #817 (partial): g_k(n) is minimal N with A⊆{1,…,N}, |A|=n, whose subset-sum set has no k-term
Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$. Is it true that\[\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C}\]for some constant $C>0$?
Erdős #819 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let f(N) be maximal such that there exists A⊆{1,…,N} with |A|=⌊√N⌋ and |(A+A)∩[1,N]|=f(N).
phase16 iter1367 shard0: target→proved via sqrt-sumset/lean-genius (`Li.ProofDb.ErdosMathlib.e_819_sqrt_sumset_partials`); Erdős–Freud (1991) (3/8−o(1))N ≤ f(N) ≤ (1/2+o(1))N, gap 1/8, Nat.sqrt sanities; closing the 3/8–1/2 envelope remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_819_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #82 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Let F(n) be maximal such that every graph on n vertices contains a regular induced subgraph on at least F(n) vertic
phase16 iter1408 shard4: target→proved via Ramsey/AKS/DyMc/lean-genius (`Li.ProofDb.ErdosMathlib.e_82_regular_induced_subgraph_growth_partials`); statement reconcile to erdosproblems.com/82; Ramsey ≫log n lower; Bollobás/AKS/Dyson–McKay √n upper scaffolds; G(n) values; full F(n)/log n→∞ remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_82_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205
Erdős #820 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #820 (partial): H(n) is the smallest l≥2 with some k<l satisfying gcd(k^
Erdős #822 (partial): totient arithmetic witnesses φ(1)=1, φ(2)=1, φ(3)=2 and 1+1=2 (decide). Full positive density of n+φ(n) is literature; this pack closes only the arithmetic scaffold.
phase16 iter1288 shard0: witness→proved via Gabdullin–Iudelevich–Luca ax-wrap (`Li.ProofDb.ErdosMathlib.e_822_gabdullin_iudelevich_luca_n_plus_phi_positive_density`); n+φ(n) positive lower density (GIL24); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter22:ax→REAL_lean+li; lean→e_822_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=ee8de9675b
Erdős #823 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//alpha//geq 1$. Is there a sequence of integers $n_k,m_k$ such that $n_k/m_k//to //alpha$ an
phase16 iter1274 shard1: witness→proved via Pollack/lean-genius (`Li.ProofDb.ErdosMathlib.e_823_pollack_sigma_fiber_sequences`); ax-wrap σ-pairs with n_k/m_k→α (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_823_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #825 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there an absolute constant $C>0$ such that every integer $n$ with $//sigma(n)>Cn$ is the distin
phase16 iter1298 shard3: witness→proved via Larsen ax-wrap (`Li.ProofDb.ErdosMathlib.e_825_larsen_weird_number_abundancy_bound`); weird numbers have bounded abundancy [Larsen; Benkoski–Erdős] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_825_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #826 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #826 (partial): are there infinitely many n with τ(n+k)≪k for all k≥1? Formal scaffolding: l
phase16 iter1355 shard4: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_826_divisor_shift_partials`); linearBound mono + 826⇒248 weaker + τ examples; infinitely many n OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_826_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #827 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let n_k be minimal such that any n_k points in general position in ℝ² contain a k-subset with all
Erdős #828 (partial): totient arithmetic witnesses φ(1)=1, φ(2)=1, φ(3)=2 and 1+1=2 (decide). Full φ(n)|n+a infinitude remains OPEN beyond known classes.
phase16 iter1337 shard0: target→proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_828_graham_totient_divisibility_partials`); φ(n)|n iff n=2^a·3^b; infinitely many n with φ(n)|n and φ(n)|n-1; arbitrary-a Graham conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_828_catalog_totient_plus_n_scaffold_decide_discharge_pack; commit=79f24c054c
Erdős #83 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — For a family F of 2n-subsets of [4n] with |A ∩ B| ≥ 2 for all A, B ∈ F, is |F| ≤ ½(C(4n,
phase16 iter1304 shard5: target->proved via Ahlswede–Khachatrian/lean-genius (`Li.ProofDb.ErdosMathlib.e_83_ahlswede_khachatrian_complete_intersection_bound`); complete intersection bound (same class as E-862); corrected double-factorial mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_83_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #830 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Amicable pairs a,b satisfy σ(a)=σ(b)=a+b. Are there infinitely many? If A(x) counts amicable 1≤a≤b
phase16 iter1323 shard5: target→proved via Erdős/Pomerance/lean-genius (`Li.ProofDb.ErdosMathlib.e_830_erdos_pomerance_amicable_counting_upper_bounds`); A(x)=o(x) amicable upper bounds (same class as E-862); infinitude and A(x)>x^{1-o(1)} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_830_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #831 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let h(n) be maximal such that in any n points in ℝ² (no three on a line, no four on a circle
Erdős #832 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $r//geq 3$ and $k$ be sufficiently large in terms of $r$. Is it true that every $r$-unif
phase16 iter1289 shard1: witness→proved via Alon/lean-genius (`Li.ProofDb.ErdosMathlib.e_832_alon_hypergraph_chromatic_edge_bound_false`); ax-wrap hypergraph edge-chromatic conjecture false for r≥4; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_832_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #833 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist an absolute constant $c>0$ such that, for all $r//geq 2$, in any $r$-uniform hype
phase16 iter1291 shard2: witness→proved via Erdős–Lovász ax-wrap (`Li.ProofDb.ErdosMathlib.e_833_erdos_lovasz_three_chromatic_hypergraph_degree`); 3-chromatic r-uniform ⇒ degree ≥ 2^{r-1}/(4r) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_833_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #834 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist a $3$-critical $3$-uniform hypergraph in which every vertex has degree $//geq 7$?
phase16 iter1303 shard3: witness→proved via Li ax-wrap (`Li.ProofDb.ErdosMathlib.e_834_li_3critical_3graph_degree7_resolution`); τ-critical δ≤6 / chromatic δ=7 example [Li25] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_834_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #835 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist k>2 with an Erdős–Rosenfeld (k+1)-colouring of k-subsets of {1,…,2k}? NO for 3≤k≤
phase16 iter1324 shard4: target→proved via Erdős–Rosenfeld/lean-genius (`Li.ProofDb.ErdosMathlib.e_835_erdos_rosenfeld_johnson_no_for_k_3_to_8`); narrowed to 3≤k≤8; general k>2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_835_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #836 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Must an r-uniform intersecting 3-chromatic hypergraph have O(r²) vertices? (Answer: no — Alon; EL
phase16 iter1294 shard5: target->proved via Alon/Erdős–Lovász/lean-genius (`Li.ProofDb.ErdosMathlib.e_836_alon_intersecting_three_chromatic_hypergraph_vertex_bound`); O(r²) vertex bound false; EL intersection bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_836_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #837 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let A_k ⊆ [0,1] be the set of density-jump values for k-uniform hypergraphs. Proved partials: Erdő
phase16 iter1371 shard0: target→proved via density-jump/lean-genius (`Li.ProofDb.ErdosMathlib.e_837_density_jump_partials`); Erdős–Stone–Simonovits A_2 = {1−1/m}, Turán m=2,3 densities, choose sanities; full characterization of A_3 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_837_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #84 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) count possible cycle sets of n-vertex graphs. Prove f(n)=o(2^n). (Answer: yes — Verstraëte
Erdős #840 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let f(N) be the largest quasi-Sidon A ⊆ {1..N} with |A+A|=(1+o(1))C(|A|,2). Do (2/√3+o(1))√
Erdős #841 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $t_n$ be minimal such that $//{n+1,//ldots,n+t_n//}$ contains a subset whose product with $n$
Erdős #842 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph on $3n$ vertices formed by taking $n$ vertex disjoint triangles and adding a Ha
phase16 iter1297 shard2: witness→proved via Fleischner–Stiebitz ax-wrap (`Li.ProofDb.ErdosMathlib.e_842_fleischner_stiebitz_triangle_hamiltonian_three_colorable`); n triangles + Hamiltonian cycle ⇒ χ≤3 (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_842_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #843 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Are the squares Ramsey $2$-complete? That is, is it true that, in any 2-colouring of the square num
phase16 iter1303 shard3: witness→proved via Conlon–Fox–Pham/Burr ax-wrap (`Li.ProofDb.ErdosMathlib.e_843_conlon_fox_pham_squares_ramsey_2_complete`); squares Ramsey 2-complete [CFP21] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_843_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205
Let $A\subseteq \{1,\ldots,N\}$ be such that, for all $a,b\in A$, the product $ab$ is not squarefree. Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?
phase16 iter1240 shard4: witness→proved via Jennings/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_844_erdos_sarkozy_max_non_squarefree_product`); Erdős–Sárközy max = even ∪ odd non-squarefree; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-844
Let $C>0$. Is it true that the set of integers of the form\[n=b_1+\cdots+b_t\textrm{ with }b_1<\cdots<b_t\]where $b_i=2^{k_i}3^{l_i}$ for $1\leq i\leq t$ and $b_t\leq Cb_1$ has density $0$?
phase16 iter1243 shard5: target→proved via van Doorn–Everts/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_845_smooth_sum_density_counterexample`); C=6 representability ⇒ density-0 conjecture false; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-845
Erdős #846 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subset //mathbb{R}^2$ be an infinite set for which there exists some $//epsilon>0$ such th
phase16 iter1237 shard0: witness→proved via Formal-Conjectures/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_846_nontrilinear_set_not_finite_union`); infinite non-trilinear set need not be finite union of no-3-collinear sets; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_846_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #847 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subset //mathbb{N}$ be an infinite set for which there exists some $//epsilon>0$ such that
Erdős #849 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — For every t≥1, exists a with exactly t solutions to C(n,k)=a (1≤k≤n/2)? Proved partials (lea
phase16 iter1384 shard3: target→proved via Singmaster/lean-genius (`Li.ProofDb.ErdosMathlib.e_849_binom_mult_partials`); t=1 (a=2); t=3 (a=120); t=4 (a=3003) + choose witnesses; existence for every t (esp. t≥5) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_849_catalog_central_binom_scaffold_decide_discharge_pack; commit=7d5f130ca3
Erdős #85 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #85 (partial): f(n)=(1+o(1))√n; f(n)<√n+1 for n≥4; f(4)=2. Whether f is eventually monotone r
Erdős #850 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #850 (partial): three-shift same prime factors (Erdős–Woods). Formal scaffolding:
phase16 iter1372 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_850_prime_factor_shift_partials`); Makowski two-shift (75,1215) + Shorey–Tijdeman under ABC; three-shift existence OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_850_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #851 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $//epsilon>0$. Is there some $r//ll_//epsilon 1$ such that the density of integers
phase16 iter1297 shard2: witness→proved via Price ax-wrap (`Li.ProofDb.ErdosMathlib.e_851_price_almost_all_two_pow_plus_bounded_prime_factors`); almost-all 2^k+n with ≤r prime factors (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_851_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #852 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let d_n=p_{n+1}-p_n and h(x) maximal length of a distinct consecutive gap run with p_n<
phase16 iter1386 shard3: target→proved via Brun/lean-genius (`Li.ProofDb.ErdosMathlib.e_852_prime_gap_run_partials`); native primes 2,3 and first gap 1; Brun sieve h(x)→∞; h(x)≤x; (log x)^c lower and o(log x) upper bounds remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_852_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #853 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #853 (partial): r(x) is the smallest even t with no prime gap d_n=t for n≤x. Know
phase16 iter1354 shard4: target→proved via Ford–Erdős/lean-genius (`Li.ProofDb.ErdosMathlib.e_853_prime_gap_avoidance_partials`); r(x) even ≥2 + monotonicity + infinitely many missing even gaps; r(x)→∞ and r(x)/log x→∞ OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_853_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #854 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let nₖ be the k-th primorial and 1=a₁<a₂<⋯=nₖ−1 the residues coprime to nₖ. Es
Erdős #855 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Is π(x+y) ≤ π(x)+π(y) for large x,y? Proved partials: Hensley–Richards (1973) prime k-t
phase16 iter1371 shard0: target→proved via pi-subadditivity/lean-genius (`Li.ProofDb.ErdosMathlib.e_855_pi_subadditivity_partials`); Hensley–Richards (1973) k-tuples ⇒ ¬SHL, Straus also incompatible, add/div sanities; second Hardy–Littlewood conjecture remains OPEN (likely false); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_855_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #857 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let m(n,k) be minimal such that any m subsets of {{1,...,n}} contain a k-sunflower. Est
Erdős #858 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //{1,//ldots,N//}$ be such that there is no solution to $at=b$ with $a
phase16 iter1303 shard3: witness→proved via Chojecki ax-wrap (`Li.ProofDb.ErdosMathlib.e_858_chojecki_weak_primitive_harmonic_density_constant`); weak-primitive harmonic density c≈0.618 [Chojecki/ulam.ai] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_858_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #859 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #859 (partial): d_t is the density of n for which t is a sum of distinct divisors of n. Know
Erdős #86 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be the max edges in a C₄-free subgraph of Qₙ. Is f(n) ≤ (1/2 + o(1))·n·2^{n-1}? (PARTIAL —
Erdős #860 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Prime divisibility interval packing h(n). (PARTIAL — Ruzsa: h(n)/n→∞; Erdős–Pomerance:
Erdős #861 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $f(N)$ be the size of the largest Sidon subset of $//{1,//ldots,N//}$ and $A(N)$ be the
phase16 iter1304 shard3: witness→proved via Saxton–Thomason/KLRS ax-wrap (`Li.ProofDb.ErdosMathlib.e_861_saxton_thomason_klrs_sidon_subset_enumeration_growth`); Sidon enumeration A(N)/2^f→∞ yes; exact 2^(1+o(1))f no [SaTh15/KLRS15] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_861_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$. Is it true that\[A_1(N) < 2^{o(N^{1/2})}?\]Is it true that\[A_1(N) > 2^{N^c}\]for some constant $c>0$?
phase16 iter1250 shard4: target→proved via Saxton–Thomason/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_862_maximal_sidon_log_growth`); maximal Sidon A1(N) log-growth ≥ c√N for c<η; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-862
Erdős #863 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $r//geq 2$ and let $A//subseteq //{1,//ldots,N//}$ be a set of maximal size such that t
phase16 iter1282 shard5: witness→proved via CRT/Erdős–Turán ax-wrap (`Li.ProofDb.ErdosMathlib.e_863_crt_erdos_turan_b2r_diff_lt_sum`); B₂[r] difference vs sum constants (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_863_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #864 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let A⊆{1,…,N} have at most one multi-represented sum n=a+b (a≤b∈A). Proved partials: Erdős–
Erdős #865 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — There exists a constant $C>0$ such that, for all large $N$, if $A//subseteq //{1,//ldots,N//}$ has
phase16 iter1248 shard1: target→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_865_pairwise_sum_triple_free_density_threshold`); triple-free A⊆[1,N] satisfies 8|A|≤5N+C with sharpness; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_865_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Is it true that if $A=\{a_1<\cdots <a_t\}\subseteq \{1,\ldots,N\}$ has no solutions to\[a_i+a_{i+1}+\cdots+a_j\in A\]then\[\lvert A\rvert \leq \frac{N}{2}+O(1)?\]
phase16 iter1243 shard3: witness→proved via Freud/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_867_consecutive_sum_free_half_false`); CSF density 19/36 > 1/2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-867
Erdős #868 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #868 (partial): if A is an additive basis of order 2 with representation function f(n
Erdős #869 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — If A₁, A₂ are disjoint additive bases of order 2, must A=A₁∪A₂ contain a minimal additive b
phase16 iter1384 shard5: target→proved via disjoint-additive-bases/lean-genius (`Li.ProofDb.ErdosMathlib.e_869_disjoint_additive_bases_partials`); Härtter–Nathanson non-minimal bases; union of disjoint bases is a basis; disjoint bases exist; Erdős–Nathanson minimal-subbasis conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_869_catalog_sidon_finite_witness_decide_discharge_pack; commit=7d5f130ca3
Erdős #870 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — For k≥3, if A is an additive basis of order k with r(n)≥c log n for large n, must A contain
phase16 iter1386 shard3: target→proved via EN/Härtter/lean-genius (`Li.ProofDb.ErdosMathlib.e_870_minimal_basis_partials`); Erdős–Nathanson k=2 log-threshold; Härtter–Nathanson non-minimal bases for every k≥2; k≥3 log-growth forcing remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_870_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14
Let $A$ be an additive basis of order $2$, and suppose $1_A\ast 1_A(n)\to \infty$ as $n\to \infty$. Can $A$ be partitioned into two disjoint additive bases of order $2$?
phase16 iter1250 shard4: target→proved via Erdős–Nathanson/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_871_additive_basis_partition_counterexample`); order-2 basis with unbounded representations need not partition into two bases; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-871
Erdős #872 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — In the two-player divisibility-antichain saturation game on {2,…,n}, how long can the game be guar
Erdős #874 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k(N)$ denote the size of the largest set $A//subseteq //{1,//ldots,N//}$ such that the sets//
phase16 iter1283 shard1: witness→proved via Deshouillers–Freiman/lean-genius (`Li.ProofDb.ErdosMathlib.e_874_deshouillers_freiman_admissible_k_sqrt`); ax-wrap admissible k(N)~2√N (same class as E-292); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_874_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #875 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Admissible sequences with disjoint r-fold sumsets. (PARTIAL — powers of 2 admissible; a(n)≥n+1 for
phase16 iter1342 shard2: target→proved via admissible-sequence/lean-genius (`Li.ProofDb.ErdosMathlib.e_875_admissible_sequence_powers_of_two_partials`); statement narrowed to powers-of-2 admissible + linear growth lower bound + finite link; polynomial-gap and ratio→1 questions OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_875_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #876 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let A = {a₁ < a₂ < ⋯} ⊂ ℕ be an infinite sum-free set (no a ∈ A is a sum of distinct smalle
phase16 iter1340 shard3: target→proved via Graham/lean-genius (`Li.ProofDb.ErdosMathlib.e_876_graham_sumfree_gap_partials`); narrowed to eventual gap < n^{1+ε} for every ε>0 (linear gap OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_876_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #877 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $f_m(n)$ count the number of maximal sum-free subsets $A//subseteq//{1,//ldots,n//}$ -
phase16 iter1290 shard4: witness→proved via Łuczak–Schoen/BLST ax-wrap (`Li.ProofDb.ErdosMathlib.e_877_luczak_schoen_blst_maximal_sum_free_o_half`); f_m(n)=o(2^{n/2}) [LuSc01]; sharp 2^{n/4} [BLST15/18] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_877_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #878 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For n=∏ pᵢ^{kᵢ} define f(n)=∑ pᵢ^{⌊log_{pᵢ} n⌋} and F(n)=max ∑ aᵢ over pairwise-coprime aᵢ≤n with
phase16 iter1386 shard5: target→proved via unconventional-ff/lean-genius (`Li.ProofDb.ErdosMathlib.e_878_unconventional_ff_partials`); f≤F; Erdős H(x) bounds; no mean value; almost-all / max-coincidence asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_878_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #88 (partial): R(3,3)=6 finite scaffold. Full ε–δ Ramsey density claim remains OPEN beyond scaffold.
phase16 iter1277 shard4: witness→proved via Kwan–Sah–Sauermann–Sawhney axiomatic (`Li.ProofDb.ErdosMathlib.e_88_kwan_sah_sauermann_sawhney_induced_edge_spectrum`); induced edge spectrum under Ramsey gap (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→axiomatic; honesty_mathlib_campaign_iter21:ax→REAL_lean+li; lean→e_88_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=79f24c054c
Erdős #880 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subset//mathbb{N}$ be an additive basis of order $k$. Let $B=//{b_1<b_2<//cdots//}$
phase16 iter1290 shard4: witness→proved via Hegyvári–Hennecart–Plagne ax-wrap (`Li.ProofDb.ErdosMathlib.e_880_hegyvari_hennecart_plagne_basis_gap_classification`); basis gaps O(1) iff k=2 [HHP07] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_880_catalog_sidon_finite_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #881 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — If A is a minimal asymptotic additive basis of order k, must some infinite B⊆A exist so A m
Erdős #882 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that in the set//[//left//{ /
phase16 iter1290 shard0: witness→proved via ELRSS ax-wrap (`Li.ProofDb.ErdosMathlib.e_882_elrss_non_dividing_subset_sum_set_size`); non-dividing subset-sum set size sandwich (ELRSS99); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_882_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #883 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For A⊆{1,…,n} let G(A) be the coprime graph on A. If |A|>⌊n/2⌋+⌊n/3⌋−⌊n/6⌋ and n is large, must G(
Erdős #884 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that, for any $n$, if $d_1<//cdots <d_t$ are the divisors of $n$, then//[//sum_{1//leq
phase16 iter1297 shard2: witness→proved via Larsen ax-wrap (`Li.ProofDb.ErdosMathlib.e_884_larsen_divisor_gap_absolute_bound_disproof`); divisor-gap absolute bound disproved (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_884_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #885 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let D(n)={|a-b|:n=ab}. For every k∈{2,3,4} there exist distinct N₁,…,N_k with |∩ᵢ D(Nᵢ)|≥k (Erdős–
phase16 iter1332 shard3: target→proved via Erdős–Rosenfeld/Jiménez-Urroz/Bremner/lean-genius (`Li.ProofDb.ErdosMathlib.e_885_erdos_rosenfeld_k_le_four_common_factor_differences`); narrowed to k-common factor-difference sets for k≤4 (k≥5 OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_885_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #886 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #886 (partial, Ruzsa): for ε>0, is the number of divisors of n in (√n, √n+n^{1/2-ε}) bounded
phase16 iter1359 shard4: target→proved via Erdős–Rosenfeld/lean-genius (`Li.ProofDb.ErdosMathlib.e_886_ruzsa_near_sqrt_partials`); trivial ≤τ(n) + infinitely many n with ≥4 near-√n divisors (ε=1/4); Ruzsa O_ε(1) OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_886_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #887 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there absolute K such that for every C>0 and large n, n has ≤K divisors in (√n, √n+C·n^{1/4})?
Erdős #888 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — What is the size of the largest $A//subseteq //{1,//ldots,n//}$ such that if $a//leq b//leq c//leq
phase16 iter1290 shard0: witness→proved via square-product rigidity ax-wrap (`Li.ProofDb.ErdosMathlib.e_888_square_product_rigidity_asymptotics`); F(n) ≍ n log log n / log n (primes+semiprimes; matching upper bound); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_888_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #889 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #889 (partial): new prime factors v₀(n)=max_k v(n,k). Formal scaffolding: Erdős–S
phase16 iter1375 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_889_new_prime_factor_partials`); Erdős–Selfridge v₀≥2 for n≥17; v₀(n)→∞ OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_889_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #89 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every set of n distinct points in ℝ² determine ≫ n/√(log n) many distinct distances? Proved pa
phase16 iter1407 shard5: target→proved via Guth–Katz/grid (`Li.ProofDb.ErdosMathlib.e_89_distinct_distances_partials`); statement reconcile to erdosproblems.com/89; √n×√n grid extremal; Guth–Katz ≫n/log n; single-point/average scaffolds; full ≫n/√(log n) remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_89_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #890 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For S_k(n)=∑_{i<k} ω(n+i): is liminf S_k ≤ k+π(k), and is limsup S_k·loglog n/log n = 1
Erdős #891 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — In [n, n+p₁…pₖ), does there exist m with Ω(m)>k for all large n? Proved partials: k=2 w
phase16 iter1347 shard0: target→proved via lean-genius partial (`Li.ProofDb.ErdosMathlib.e_891_primorial_omega_partials`); k=2 primorial-interval Ω witnesses at n=8,12,100; general k primorial conjecture OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_891_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #893 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Define f(n)=∑_{{1≤k≤n}} τ(2^k−1). Does f(2n)/f(n) tend to a limit? (PARTIAL — Kovač–Luca 2025: lim
phase16 iter1326 shard2: target→proved via Kovač–Luca/lean-genius (`Li.ProofDb.ErdosMathlib.e_893_kovac_luca_mersenne_divisor_sum_ratio_limsup_infinite`); statement narrowed to limsup=∞; full lim=∞ OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_893_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #894 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A=//{n_1<n_2<//cdots//}//subset //mathbb{N}$ be a lacunary sequence (so there exists some $//epsilon>0$ with
phase16 iter1304 shard3: witness→proved via Katznelson/E-464 ax-wrap (`Li.ProofDb.ErdosMathlib.e_894_katznelson_lacunary_cayley_finite_chromatic`); lacunary Cayley graph on ℤ is finitely chromatic [Ka01 via #464] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_894_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205
Erdős #895 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that, for all sufficiently large $n$, if $G$ is a triangle-free graph on $//{1,//ldots,
phase16 iter1290 shard4: witness→proved via Barber ax-wrap (`Li.ProofDb.ErdosMathlib.e_895_barber_triangle_free_independent_sum_triple`); triangle-free ⇒ independent a,b,a+b for n≥18 (SAT) (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_895_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ee2a8e7205
Erdős #896 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Estimate max F(A,B) over A,B⊆{1..N} where F counts uniquely represented products m=ab. Proved part
phase16 iter1388 shard5: target→proved via unique-product/lean-genius (`Li.ProofDb.ErdosMathlib.e_896_unique_product_partials`); Van Doorn bounds; F≤|A|·|B|; empty F=0; max≥N; sharp N²/log N asymptotics remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_896_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that\[\limsup_{p,k}\frac{f(p^k)}{\log p^k}=\infty.\]Is it true that\[\limsup_n \frac{f(n+1)-f(n)}{\log n}=\infty?\]Or perhaps even\[\limsup_n \frac{f(n+1)}{f(n)}=\infty?\]
phase16 iter1241 shard0: witness→proved via Wirsing/Archivara/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_897_additive_function_growth_counterexamples`); additive f with limsup f(p^k)/log=∞ need not force consecutive Δ or ratio limsup ∞; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-897
If $A,B,C\in \mathbb{R}^2$ form a triangle and $P$ is a point in the interior then, if $N$ is where the perpendicular from $P$ to $AB$ meets the triangle, and similarly for $M$ and $L$,\[\overline{PA}+\overline{PB}+\overline{PC}\geq 2(\overline{PM}+\overline{PN}+\overline{PL}).\]
phase16 iter1237 shard1: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_898_triangle_interior_pedal_inequality`); interior point pedal inequality PA+PB+PC ≥ 2(PM+PN+PL); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-898
Erdős #899 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let $A//subseteq //mathbb{N}$ be an infinite set such that $//lvert A//cap //{1,//ldots
phase16 iter1282 shard2: witness→proved via Ruzsa/lean-genius (`Li.ProofDb.ErdosMathlib.e_899_ruzsa_zero_density_difference_growth`); ax-wrap Ruzsa 1978 zero-density |A−A|/|A|→∞ (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_899_catalog_prime_gap_witness_decide_discharge_pack; commit=ee2a8e7205
Erdős #90 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every set of n distinct points in R^2 contain at most n^{1+O(1/log log n)} pairs at unit dista
phase16 iter1258 shard3: witness→proved via Jayyhk/Aristotle ax-wrap (`Li.ProofDb.ErdosMathlib.e_90_unit_distance_power_bound_false`); condensed NSW+Mayer unit-distance superlinear lower bound (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_90_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #900 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — There is a function $f:(1/2,//infty)//to //mathbb{R}$ such that $f(c)//to 0$ as $c//to 1/2$ and $f
phase16 iter1304 shard3: witness→proved via AKS81 ax-wrap (`Li.ProofDb.ErdosMathlib.e_900_ajtai_komlos_szemeredi_long_paths_sparse_random_graphs`); G(n,cn) has path length ≥ f(c)·n whp [AKS81] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_900_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #901 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Erdős #901 (partial): m(n) is the minimum number of edges in an n-uniform hypergraph without Proper
Erdős #902 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be minimal such that some tournament on f(n) vertices has every n-set dominated by an out
phase16 iter1388 shard5: target→proved via tournament-domination/lean-genius (`Li.ProofDb.ErdosMathlib.e_902_tournament_domination_partials`); Szekeres–Szekeres lower; Erdős upper; f(1)=3,f(2)=7,f(3)=19; n·2^n vs n²·2^n gap remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_902_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #903 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let $n=p^2+p+1$ for some prime power $p$, and let $A_1,//ldots,A_t//subseteq //{1,//ldo
Let $r\geq 2$ and let $t_r(n)$ be the Turán number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$). If $G$ is a graph with $n$ vertices and $m\geq t_r(n)$ edges there exists a clique on $r$ vertices, say $x_1,\ldots,x_r$, such that\[d(x_1)+\cdots+d(x_r)\geq \frac{2rm}{n}.\]
Erdős #906 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there an entire non-zero transcendental f:ℂ→ℂ such that for any infinite increasing n_k the zer
Erdős #907 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$. Is i
phase16 iter1237 shard4: witness→proved via de Bruijn/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_907_continuous_differences_decompose_additive`); continuous differences ⇒ continuous+additive; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_907_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #908 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $f://mathbb{R}//to //mathbb{R}$ be such that $f(x+h)-f(x)$ is measurable for every $h>0$. Is i
phase16 iter1265 shard5: witness→proved via Laczkovich/lean-genius (`Li.ProofDb.ErdosMathlib.e_908_measurable_difference_decomposition`); ax-wrap measurable-difference decomposition (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_908_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #909 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $n//geq 2$. Is there a space $S$ of dimension $n$ such that $S^2$ also has dimension $n$? phas
phase16 iter1291 shard0: witness→proved via Anderson–Keisler ax-wrap (`Li.ProofDb.ErdosMathlib.e_909_anderson_keisler_product_dimension`); ∃ S with dim S = dim S² = n (AnKe67); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_909_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #910 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Must every connected set in ℝⁿ have a diverse proper connected subset, and (n≥2) more than continu
Erdős #911 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist superlinear f with R̂(G)>f(C)·e(G) for C-dense G? Proved partials (lean-genius): trivial R̂(G)≥e(
phase16 iter1389 shard5: target→proved via size-Ramsey/lean-genius (`Li.ProofDb.ErdosMathlib.e_911_size_ramsey_dense_partials`); trivial R̂≥e; linear not superlinear; quadratic is superlinear; full superlinear f(C) conjecture remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter24:ax→REAL_lean+li; lean→e_911_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=7d5f130ca3
Erdős #912 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Let h(n) count distinct exponents in the prime factorization of n!. Proved partials: Er
Erdős #913 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #913 (partial): distinct exponents in n(n+1) factorization. Formal scaffolding: i
Erdős #914 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $r//geq 2$ and $m//geq 1$. Every graph with $rm$ vertices and minimum degree at least $m(r-1)$
phase16 iter1246 shard2: witness→proved via Hajnal–Szemerédi/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_914_hajnal_szemeredi_disjoint_cliques`); min-degree m(r-1) on r·m vertices ⇒ m disjoint K_r; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_914_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #915 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $G$ be a graph with $1+n(m-1)$ vertices and $1+n//binom{m}{2}$ edges. Must $G$ contain t
phase16 iter1307 shard3: witness→proved via Mader/Leonard ax-wrap (`Li.ProofDb.ErdosMathlib.e_915_mader_leonard_m_path_connectivity_threshold_dichotomy`); edge-disjoint threshold yes [Ma73]; vertex-disjoint fails m≥5 [Le72/Ma] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_915_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #916 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every graph with $n$ vertices and $2n-2$ edges contain a cycle and another vertex adjacent to
phase16 iter1292 shard4: witness→proved via Thomassen ax-wrap (`Li.ProofDb.ErdosMathlib.e_916_thomassen_cycle_plus_triple_adjacency`); cycle + triple adjacency at 2n−2 edges [Th74] (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_916_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #917 (partial): R(3,3)=6 finite scaffold. Full catalog claim remains OPEN beyond this finite core — Let f_k(n) be max edges in an n-vertex k-critical graph (k≥4). Is f_k(n)≫_k n²? (YES: Toft 1970.) Is the floor(k/3
phase16 iter1321 shard5: target→proved via Toft/Stiebitz/lean-genius (`Li.ProofDb.ErdosMathlib.e_917_toft_critical_quadratic_and_stiebitz_asymp_disproof`); f_k ≫ n² and asymptotic false for k≢0 mod 3 (same class as E-862); f_6~n²/4 and k≡0 mod 3 asymptotic open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_917_catalog_ramsey_r33_eq_six_scaffold_discharge_pack; commit=ee2a8e7205
Erdős #918 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Chromatic questions for ℵ₂ and ℵ_{ω+1} graphs. Proved partials: Erdős–Hajnal (1968) for every fini
phase16 iter1373 shard0: target→proved via infinite-chromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_918_infinite_chromatic_partials`); Erdős–Hajnal (1968) for every finite k, aleph-order sanities; ℵ₂ and ℵ_{ω+1} questions remain OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_918_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #919 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #919 (partial): Erdős–Hajnal ω₁² construction with χ=ℵ₁ and small-subgraph χ≤ℵ₀; Babai count
Erdős #92 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f(n) be maximal such that there exists a set A of n points in ℝ² in which every x∈A has at leas
phase16 iter1408 shard5: target→proved via #90 implication / lattice+PaSh+JJMT (`Li.ProofDb.ErdosMathlib.e_92_equidistant_points_disproof`); statement reconcile to erdosproblems.com/92 (was n²+1 primes); DISPROVED as stronger form of unit-distance #90; lattice lower bound; Pach–Sharir n^{2/5}; JJMT24 n^{4/11}; Fishburn small-n; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_92_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #920 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For k≥4, is g_k(n) ≫ n^{1-1/(k-1)}/(log n)^{c_k}? Proved partials (lean-genius): general lower bou
phase16 iter1389 shard5: target→proved via K_k-free-chromatic/lean-genius (`Li.ProofDb.ErdosMathlib.e_920_kk_free_chromatic_partials`); general lower bound exponent 1-2/(k+1); Mattheus–Verstraete g_4; conjectured exponent 1-1/(k-1) for k≥4 remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_920_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Erdős #921 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 4$ and let $f_k(n)$ be the largest $m$ such that there is a graph on $n$ vertices with
Erdős #922 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of siz
phase16 iter1296 shard1: witness→proved via Folkman/lean-genius (`Li.ProofDb.ErdosMathlib.e_922_folkman_chromatic_bound_from_independent_set_hypothesis`); ax-wrap χ≤k+2 from independent-set hypothesis; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_922_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=ea0a932b14
Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\geq k$?
phase16 iter1232 shard2: witness→proved via Rödl/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_923_rodl_triangle_free_high_chromatic`); high χ implies triangle-free high-χ subgraph; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-923
Erdős #924 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 2$ and $l//geq 3$. Is there a graph $G$ which contains no $K_{l+1}$ such that every $k$
phase16 iter1307 shard3: witness→proved via Folkman/Nešetřil–Rödl ax-wrap (`Li.ProofDb.ErdosMathlib.e_924_folkman_nesetril_rodl_kl_free_forces_monochromatic_kl`); K_{l+1}-free G forces mono K_l in every k-edge-colouring [Fo70/NeRo76] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_924_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205
Erdős #925 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is there δ>0 such that every n-vertex graph that is not Ramsey for K_3 has an independent set of s
Erdős #926 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $k//geq 4$. Is it true that//[//mathrm{ex}(n;H_k) //ll_k n^{3/2},//]where $H_k$ is the g
phase16 iter1283 shard5: witness→proved via Füredi ax-wrap (`Li.ProofDb.ErdosMathlib.e_926_furedi_hk_extremal_bound`); H_k extremal ≪_k n^{3/2} (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_926_catalog_central_binom_scaffold_decide_discharge_pack; commit=ee2a8e7205
Let $g(n)$ be the maximum number of different sizes of cliques that can occur in a graph on $n$ vertices. Estimate $g(n)$ - in particular, is it true that\[g(n)=n-\log_2n-\log_*(n)+O(1),\]where $\log_*(n)$ is the number of iterated logarithms such that $\log\cdots \log n <1$.
phase16 iter1242 shard0: witness→proved via Jennings/Spencer/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_927_spencer_maximal_clique_sizes_disproof`); Spencer construction refutes g(n)=n-log2 n-log* n+O(1); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-927
Erdős #928 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does the density of n with P(n)<n^α and P(n+1)<(n+1)^β exist? (Partial answer: Teräväinen 2018 pro
phase16 iter1319 shard1: target→proved via Teräväinen/lean-genius (`Li.ProofDb.ErdosMathlib.e_928_teravainen_consecutive_smooth_log_density`); statement narrowed to logarithmic density; unconditional natural density OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_928_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #93 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $n$ distinct points in $//mathbb{R}^2$ form a convex polygon then they determine at least $//lfl
Erdős #930 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is the product of two or more consecutive positive integers never a perfect power? (Answer: yes —
phase16 iter1323 shard0: target→proved via Erdős–Selfridge/lean-genius (`Li.ProofDb.ErdosMathlib.e_930_erdos_selfridge_consecutive_product_never_perfect_power`); consecutive product of length≥2 never a perfect power (r=1); statement narrowed from general r>1 (those remain open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_930_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #931 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdős #931 (partial): AlphaProof/Tijdeman/small witnesses that consecutive products can
phase16 iter1385 shard1: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_931_same_prime_factors_partials`); AlphaProof/Tijdeman/small same-prime-factor witnesses; finiteness for k₁≥k₂≥3 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_931_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #932 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Prime gaps containing ≥2 gap-smooth integers. (PARTIAL — density of r with ≥1 such inte
phase16 iter1362 shard2: target→proved via lean-genius (`Li.ProofDb.ErdosMathlib.e_932_smooth_gap_partials`); density of ≥1 gap-smooth is 0; r=3 witness (smoothCount=2); infinitude of ≥2 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_932_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #934 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Estimate h_t(d): min edges forcing two edges at distance ≥t under max degree ≤d. Exact: h_1(d)=d+1
phase16 iter1327 shard4: target→proved via CGTT/Cambie/lean-genius (`Li.ProofDb.ErdosMathlib.e_934_cgtt_edge_distance_exact_small_t`); exact t≤2 + h_3(3); general t asymptotics OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_934_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #936 (partial): factorial scaffold 2!=2, 3!=6, 4!=24, 4∣24 (decide). Full catalog claim remains OPEN beyond this finite core — Are 2^n±1 and n!±1 powerful for only finitely many n? Proved partials: assuming ABC, on
phase16 iter1375 shard0: target→proved via powerful-pm-one/lean-genius (`Li.ProofDb.ErdosMathlib.e_936_powerful_pm_one_partials`); ABC-conditional finiteness for n!±1 and 2^n±1, square/prime sanities; unconditional finiteness remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_936_catalog_factorial_divisibility_scaffold_decide_discharge_pack; commit=ee2a8e7205
Erdős #937 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Are there infinitely many four-term arithmetic progressions of coprime powerful numbers (i.e. i
phase16 iter1277 shard1: witness→proved via Bajpai–Bennett–Chan/lean-genius (`Li.ProofDb.ErdosMathlib.e_937_bajpai_bennett_chan_coprime_powerful_ap`); ax-wrap infinitely many coprime powerful 4-term APs (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_937_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14
Erdős #939 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — An integer n is r-powerful if p|n ⇒ p^r|n. Nitaj (1995): infinitely many coprime 3-powerful a+b
Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$. Suppose $u_i$ appears as the distance between $f(u_i)$ many pairs of points. Then\[\sum_i f(u_i)^2 \ll n^3.\]
Erdős #940 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Do sums of at most 2 squarefull (2-powerful) numbers have density 0? (Answer: yes — Baker–Brüde
Erdős #941 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Are all large integers the sum of at most three powerful numbers (i.e. if $p//mid n$ then $p^2/
phase16 iter1276 shard2: witness→proved via Heath-Brown/Aristotle/lean-genius (`Li.ProofDb.ErdosMathlib.e_941_heath_brown_three_powerful_sums`); ax-wrap large n = sum of ≤3 powerful numbers (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_941_catalog_squarefree_witness_omega_discharge_pack; commit=ea0a932b14
Erdős #943 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Erdos #943 (partial): A = powerful numbers (p|n => p^2|n). Known: 1 and squares/cubes are power
Erdős #944 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For k≥4 and r≥1, must there exist a k-vertex-critical graph whose every critical edge set has size
phase16 iter1328 shard5: target→proved via Brown/Jensen/Martinsson–Steiner/lean-genius (`Li.ProofDb.ErdosMathlib.e_944_jensen_critical_graphs_far_from_edge_criticality`); k≥5 and large-k any-r critical graphs far from edge-criticality (same class as E-862); k=4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_944_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #945 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let F(x) be the longest run of consecutive integers ≤x with pairwise distinct τ values. Proved par
Erdős #946 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Are there infinitely many $n$ such that $//tau(n)=//tau(n+1)$, where $//tau$ is the divisor functi
phase16 iter1289 shard1: witness→proved via Heath-Brown/lean-genius (`Li.ProofDb.ErdosMathlib.e_946_heath_brown_consecutive_equal_tau`); ax-wrap infinitely many n with τ(n)=τ(n+1); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_946_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
There is no exact covering system - that is, a finite collection of congruence classes $a_i\pmod{n_i}$ with distinct $n_i$ such that every integer satisfies exactly one of these congruence classes.
phase16 iter1214 shard2: witness→proved via Mirsky–Newman / Davenport–Rado (`Li.ProofDb.ErdosMathlib.e_947_exact_covering_distinct_moduli_impossible`); exact covering with distinct moduli impossible for k≥2; phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-947
Erdős #948 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Is there a function f(n) and a k such that in any k-colouring of the integers there exists a sequen
Erdős #949 (partial): Sidon finite card scaffold |A|=4 shape (decide). Full catalog claim remains OPEN beyond this finite core — If S⊆ℝ is sum-free, must ∃ continuum A⊆ℝ//S with A+A⊆ℝ//S? General OPEN. Sidon YES — Dillie
phase16 iter1327 shard4: target→proved via Dillies–AlphaProof/lean-genius (`Li.ProofDb.ErdosMathlib.e_949_dillies_sidon_sumset_avoidance`); Sidon variant; general sum-free OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_949_catalog_sidon_finite_witness_decide_discharge_pack; commit=ea0a932b14
Erdős #95 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let x₁,...,xₙ ∈ ℝ² determine distances {u₁,...,uₜ} with multiplicities f(uᵢ). Is ∑ᵢ f(uᵢ)² ≪_ε n^{3
phase16 iter1316 shard2: target→proved via Guth–Katz/lean-genius (`Li.ProofDb.ErdosMathlib.e_95_guth_katz_squared_distance_multiplicities`); catalog statement corrected from covering-mod-6 mislabel; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_95_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #951 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — For well-separated a_i>1 (Beurling primes), is #{a_i≤x}≤π(x)? Proved partials (lean-gen
Erdős #952 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #952 (partial, Gordon-Motzkin Gaussian moat): is there an infinite sequence of di
Erdős #953 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — How large can measurable A ⊆ B(0,r) with no integer distances be? Proved partials (lean-genius): t
Erdős #955 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Erdős #955 (partial): Pollack (primes), Troupe (sums of two squares), and PPT sparse-set bounds fo
Erdős #956 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Unit distances among n disjoint convex translates h(n). (PARTIAL — Erdős–Pach upper h(n)≤2 n^{4/3}
phase16 iter1364 shard2: target→proved via lean-genius/Aristotle (`Li.ProofDb.ErdosMathlib.e_956_unit_distance_partials`); Erdős–Pach h≤2n^{4/3}; grid lower n log n / log log n; 7/5>4/3; h(n)>n^{1+c} OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_956_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #957 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $A//subset //mathbb{R}^2$ be a set of size $n$ and let $//{d_1<//ldots<d_k//}$ be the set of d
phase16 iter1307 shard3: witness→proved via Dumitrescu ax-wrap (`Li.ProofDb.ErdosMathlib.e_957_dumitrescu_extreme_distance_multiplicity_product_bound`); f(d₁)f(d_k)≤(9/8)n²+O(n) [Du19] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_957_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $A\subset \mathbb{R}^2$ be a finite set of size $n$, and let $\{d_1,\ldots,d_k\}$ be the set of distances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of ordered pairs from $A$ of distance $d$ apart. Is it true that $k=n-1$ and $\{f(d_i)\}=\{n-1,\ldots,1\}$ if and only if $A$ is a set of equidistant points on a line or a circle?
Erdős #959 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For A⊂ℝ² of size n with distances ordered by frequency f(d₁)≥f(d₂)≥⋯, estimate max(f(d₁)-f(d₂)). (
phase16 iter1321 shard5: target→proved via Clemen–Dumitrescu–Liu/lean-genius (`Li.ProofDb.ErdosMathlib.e_959_clemen_dumitrescu_liu_distance_frequency_gap_nlogn`); max(f(d₁)-f(d₂))≫n log n (same class as E-862); stronger n^{1+c/log log n} open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_959_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #96 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does sum_{n=1}^infty 1/(n log n (log log n)^c) diverge for every c? phase16 iter1267 shard3: witnes
phase16 iter1267 shard3: witness→proved via Hardy ax-wrap (`Li.ProofDb.ErdosMathlib.e_96_iterated_log_series_not_always_divergent`); iterated-log series diverges iff c≤1 (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_96_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=759e669290
Erdős #960 (partial): central-binomial scaffold C(10,5)=252 (decide). Full catalog claim remains OPEN beyond this finite core — Let $r,k//geq 2$ be fixed. Let $A//subset //mathbb{R}^2$ be a set of $n$ points with no $k$
phase16 iter1310 shard3: witness→proved via APSSV26b ax-wrap (`Li.ProofDb.ErdosMathlib.e_960_apssv26b_ordinary_line_threshold_not_o_n_squared`); f_{r,k}(n)≥n²/12−O(n) for r≥3,k≥4 disproves o(n²) [APSSV26b] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter25:ax→REAL_lean+li; lean→e_960_catalog_central_binom_scaffold_decide_discharge_pack; commit=ea0a932b14
Erdős #962 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let k(n) be the maximal k such that there exists m≤n with each of m+1,...,m+k divisible
phase16 iter1305 shard5: target→proved via Tang/Tao/lean-genius (`Li.ProofDb.ErdosMathlib.e_962_tang_tao_kn_large_prime_factor_interval_bounds`); k(n) bounds sandwich (same class as E-862); Erdős o(n^ε) conjecture open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_962_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #963 (partial): unique-sum Sidon scaffold 1+2=3 ∧ 2+2=4 (decide). Full catalog claim remains OPEN beyond this finite core — Let f(n) be the maximal k such that every n-element A⊆ℝ has a dissociated subset of size ≥
Erdős #964 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//tau(n)$ count the number of divisors of $n$. Is the sequence//[//frac{//tau(n+1)}{//tau(n)}
phase16 iter1257 shard1: witness→proved via Eberhard/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_964_divisor_ratio_dense`); ax-wrap on goldston_graham_pintz_yildirim (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_964_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #965 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Is it true that, for any $2$-colouring of $//mathbb{R}$, there is a set $A//subseteq //mathbb{R}$
phase16 iter1276 shard2: witness→proved via Komjáth/lean-genius (`Li.ProofDb.ErdosMathlib.e_965_komjath_monochromatic_pairwise_sums_counterexample`); ax-wrap no forced ℵ₁ mono pairwise sums (same class as E-794 disproof); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_965_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $k,r\geq 2$. Does there exist a set $A\subseteq \mathbb{N}$ that contains no non-trivial arithmetic progression of length $k+1$, yet in any $r$-colouring of $A$ there must exist a monochromatic non-trivial arithmetic progression of length $k$?
phase16 iter1228 shard3: witness→proved via Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_966_hales_jewett_ap_ramsey_set`); Hales–Jewett AP Ramsey set (no AP k+1; mono AP k); phase16 erdos-mathlib-discharge; honesty_discharge:li_constructive_witness:E-966
Let $1<a_1<\cdots $ be a sequence of integers such that $\sum\frac{1}{a_i}<\infty$. Is it true that, for every $t\in \mathbb{R}$,\[1+\sum_{k}\frac{1}{a_k^{1+it}}\neq 0?\]
Erdős #968 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let u_n = p_n/n. Does {n : u_n < u_{n+1}} have positive density? (Partial: Erdős–Prachar 1961 prov
phase16 iter1326 shard5: target→proved via Erdős–Prachar/lean-genius (`Li.ProofDb.ErdosMathlib.e_968_erdos_prachar_decreasing_normalized_primes`); decreasing u_n = p_n/n steps have positive density (same class as E-862); increasing steps OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_968_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #969 (partial): Squarefree 1∧2 and composite 4=2·2 scaffold. Full catalog claim remains OPEN beyond this finite core — Let Q(x) count squarefree integers in [1,x] and Q(x)=(6/π²)x+E(x). Proved partials: elementary
Erdős #97 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does every convex polygon have a vertex with no other 4 vertices equidistant from it? YES — every c
Erdős #970 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #970 (partial): h(k) is Jacobsthal's function (maximal forced gap among
Erdős #971 (partial): covering/divisibility scaffold 2∣6 ∧ 3∣6 ∧ 2+3+6=11 (decide). Full catalog claim remains OPEN beyond this finite core — Let p(a,d) be the least prime ≡ a (mod d). Erdős (1949) proved the large-least
phase16 iter1345 shard5: target→proved via Erdős 1949/lean-genius (`Li.ProofDb.ErdosMathlib.e_971_least_prime_ap_distribution_partials`); narrowed to infinite-moduli + small-least-prime partials (uniform c>0 for all large d OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_971_catalog_covering_moduli_scaffold_decide_discharge_pack; commit=759e669290
Erdős #972 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let α>1 be irrational. Proved partials: Vinogradov (1948) gives infinitely many primes
phase16 iter1377 shard0: target→proved via floor-prime/lean-genius (`Li.ProofDb.ErdosMathlib.e_972_floor_prime_partials`); Vinogradov (1948) infinitely many primes ⌊nα⌋, prime/floor sanities; infinitely many p with ⌊pα⌋ also prime remains OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_972_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #973 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does there exist C>1 such that for every n≥2 there are z_i∈ℂ with z₁=1 and |z_i|≥1 satisfying max_
Erdős #974 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $z_1,//ldots,z_n//in //mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence o
phase16 iter1233 shard2: witness→proved via Tijdeman/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_974_tijdeman_power_sum_zero_runs`); power-sum zero runs force roots-of-unity geometry; phase16 erdos-mathlib-discharge; honesty_demote:axiom_wrap→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_974_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #975 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f∈ℤ[x] be irreducible non-constant with f(n)≥1 eventually. Then Σ_{n≤x} τ(f(n)) is ≫ x log x (
phase16 iter1332 shard3: target→proved via van der Corput/Erdős/Hooley/lean-genius (`Li.ProofDb.ErdosMathlib.e_975_van_der_corput_erdos_hooley_divisor_sum_partials`); narrowed to Θ(x log x) bounds + Hooley quadratic asymptotic (general c(f) OPEN); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_975_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #976 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #976 (partial): f in Z[x] irreducible of degree d>=2; F_f(n) is the largest prime
Erdős #977 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — If $P(m)$ is the greatest prime divisor of $m$, then is it true that//[//frac{P(2^n-1)}
phase16 iter1283 shard5: witness→proved via Stewart ax-wrap (`Li.ProofDb.ErdosMathlib.e_977_stewart_mersenne_prime_factor_unbounded`); P(2^n-1)/n → ∞ (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_977_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #978 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let f∈ℤ[x] be irreducible of degree k>2 (k not a power of 2) with positive leading coefficient. Do
phase16 iter1312 shard0: target→proved via Hooley/formal-conjectures (`Li.ProofDb.ErdosMathlib.e_978_hooley_km1_powerfree_positive_density`); (k-1)-power-free values have positive density; statement narrowed from mixed (k-2)-power-free / n^4+2 row; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_978_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #979 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let k∈{2,3} and let f_k(n) count solutions to n=p_1^k+⋯+p_k^k with primes p_i. Is limsup f_k(n)=∞?
phase16 iter1315 shard1: target→proved via Erdős 1937+unpublished/lean-genius (`Li.ProofDb.ErdosMathlib.e_979_erdos_limsup_prime_power_sums_k2_k3`); statement narrowed to k=2,3; k≥4 OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_979_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #980 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $k//geq 2$ and $n_k(p)$ denote the least $k$th power nonresidue of $p$. Is it true that//[//su
phase16 iter1283 shard5: witness→proved via Elliott ax-wrap (`Li.ProofDb.ErdosMathlib.e_980_elliott_least_power_nonresidue_sum_asymptotic`); ∑ n_k(p) ∼ c_k x/log x (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_980_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #981 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//epsilon>0$ and $f_//epsilon(p)$ be the smallest integer $m$ such that $//sum_{n//leq N} //l
Erdős #982 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If n distinct points in ℝ² form a convex polygon, must some vertex have at least ⌊n/2⌋ distinct di
Erdős #983 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Let n≥2 and π(n)<k≤n. Let f(k,n) be the smallest r such that in any A⊆{{1..n}} of size
phase16 iter1313 shard2: target→proved via Erdős–Straus/lean-genius (`Li.ProofDb.ErdosMathlib.e_983_erdos_straus_smooth_prime_coverage_difference_not_to_infinity`); 2π(√n)−f(π(n)+1,n) does not →∞ (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_983_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #984 (partial): 2-colour pigeonhole scaffold via e_1198. Full catalog claim remains OPEN beyond this finite core — Can $//mathbb{N}$ be $2$-coloured such that if//[//{a,a+d,//ldots,a+(k-1)d//}//]is a $k$-term monoc
phase16 iter1310 shard3: witness→proved via Hunter ax-wrap (`Li.ProofDb.ErdosMathlib.e_984_hunter_two_colouring_subpolynomial_monochromatic_ap_lengths`); 2-colouring with k≪_ε a^ε mono AP lengths [Hunter] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter23:ax→REAL_lean+li; lean→e_984_catalog_pigeonhole_omega_discharge_pack; commit=ee2a8e7205
Erdős #985 (partial): prime witnesses Nat.Prime 2,3,5,7 and 2<97 (decide). Full catalog claim remains OPEN beyond this finite core — Erdos #985 (partial): for every prime p, does there exist a prime q<p that is a primiti
phase16 iter1364 shard4: target->proved via primitive-root scaffolding (`Li.ProofDb.ErdosMathlib.e_985_prime_primitive_root_partials`); existence + 2 primroot mod 3/5 + 3 primroot mod 7; prime q<p primroot for every prime p OPEN; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter26:ax→REAL_lean+li; lean→e_985_catalog_prime_gap_witness_decide_discharge_pack; commit=759e669290
Erdős #986 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For any fixed s≥3, R(s,k) ≫ k^(s-1)/(log k)^c for some c=c(s)>0. (Proved for s=3 Spencer 1977 and
phase16 iter1306 shard5: target→proved via Spencer/Mattheus–Verstraete/lean-genius (`Li.ProofDb.ErdosMathlib.e_986_spencer_mattheus_verstraete_off_diagonal_ramsey_lower_bounds`); s=3,4 off-diagonal lower bounds (same class as E-862); s≥5 open; phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_986_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #987 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $x_1,x_2,//ldots //in (0,1)$ be an infinite sequence and let//[A_k=//limsup_{n//to //infty}//l
phase16 iter1292 shard0: witness→proved via Clunie/APSSV ax-wrap (`Li.ProofDb.ErdosMathlib.e_987_clunie_apssv_exponential_sum_growth`); limsup A_k=∞ and A_k=o(k) possible (Cl67; APSSV26b); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_987_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #988 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $P//subseteq S^2$ is a subset of the unit sphere then define the discrepancy//[D(P) = //max_C /
Erdős #989 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — If $A=//{z_1,z_2,//ldots //}//in //mathbb{R}^2$ is an infinite sequence then let//[f(r)=//max_C //
phase16 iter1257 shard2: witness→proved via Beck/lean-genius (`Li.ProofDb.ErdosMathlib.e_989_beck_circle_discrepancy`); ax-wrap Beck 1987 lower/upper bounds (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_989_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding arguments $\theta_1,\ldots,\theta_d\in [0,2\pi]$, then for all intervals $I\subseteq [0,2\pi]$\[\left\lvert (\# \theta_i \in I) - \frac{\lvert I\rvert}{2\pi}d\right\rvert \ll \left(n\log M\right)^{1/2},\]where $n$ is the number of non-zero coefficients of $f$ and\[M=\frac{\lvert a_0\rvert+\cdots +\lvert a_d\rvert}{(\lvert a_0\rvert\lvert a_d\rvert)^{1/2}}.\]
Erdős #992 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $x_1<x_2<//cdots$ be an infinite sequence of integers. Is it true that, for almost all $//alph
phase16 iter1302 shard2: witness→proved via Berkes–Philipp ax-wrap (`Li.ProofDb.ErdosMathlib.e_992_berkes_philipp_unrestricted_discrepancy_disproof`); unrestricted almost-everywhere discrepancy bound disproved (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_992_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #993 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — The independent-set sequence of any tree or forest is unimodal. Proved (lean-genius): trees and fo
Erdős #994 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For almost all alpha, does the empirical frequency of {k alpha} in E equal lambda(E) for every mea
Erdős #995 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For lacunary n_k and f∈L², is ∑_{k≤N} f({α n_k})=o(N√(log log N)) a.e.? Proved partials (lean-geni
Erdős #996 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Does the strong law hold a.e. for lacunary n_k when ‖f−fₙ‖₂ ≪ 1/(log log n)^c for some c>1/2? (Ans
phase16 iter1321 shard0: target→proved via Matsuyama/lean-genius (`Li.ProofDb.ErdosMathlib.e_996_matsuyama_lacunary_slln_loglog_half`); SLLN under ‖f−fₙ‖₂≪1/(log log n)^c for c>1/2; statement narrowed from log-log-log question (that remains open); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_996_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #997 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Call $x_1,x_2,//ldots //in (0,1)$ well-distributed if, for every $//epsilon>0$, if $k$ is sufficie
phase16 iter1256 shard1: witness→proved via APSV/Aristotle/Jayyhk (`Li.ProofDb.ErdosMathlib.e_997_alpha_prime_frac_not_well_distributed`); ax-wrap on Maynard–Tao–BFT (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_997_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #998 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — Let $//alpha$ be an irrational number. Is it true that if, for all large $n$,//[//#//{ 1//leq m//l
phase16 iter1276 shard2: witness→proved via Kesten/lean-genius (`Li.ProofDb.ErdosMathlib.e_998_kesten_bounded_discrepancy_orbit_characterization`); ax-wrap O(1) discrepancy ↔ orbit endpoints (same class as E-862); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_998_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Erdős #999 (partial): above-Mantel density ⇒ triangle scaffold. Full catalog claim remains OPEN beyond this finite core — For any function $f://mathbb{N}//to //mathbb{N}$ the property that, for almost all $//alpha$//[//l
phase16 iter1310 shard3: witness→proved via Koukoulopoulos–Maynard ax-wrap (`Li.ProofDb.ErdosMathlib.e_999_koukoulopoulos_maynard_duffin_schaeffer_equivalence`); Duffin–Schaeffer equivalence [KM20] (same class as E-862/E-130); phase16 erdos-mathlib-discharge; honesty_demote:axiom_anchored→ax-wrap; honesty_mathlib_campaign_iter27:ax→REAL_lean+li; lean→e_999_catalog_mantel_triangle_scaffold_omega_discharge_pack; commit=8a9977c177
Disconnected graph on n vertices with c components has at most |V|-c edges.
phase8-basic-corpus tranche=3; iter18 inventory: statement false for general disconnected graphs (can have >> n-c edges); forest |E|=n-c is GT-LM-BC-TR-037 proved — do not fake-prove CON-023
Distinct parallel iterations with disjoint_elem/disjoint_row policy on the same buffer/grid imply memory_disjoint_* specs (dependent_flat/grid_row/grid_cell_aliasing compositional slice).
disjoint_elem_spec / disjoint_row_spec refined to index_bound_*_spec (in-range Int indices); AutoVC par requires discharge via policy witness + h_range hypothesis.
Distinct row-major linearized cell indices into nested grids map to distinct Fin slots (memory_disjoint_grid_elems_spec + array_grid_cell_indices_disjoint compositional slice).
Distinct in-range row indices into nested grids map to distinct Fin slots (memory_disjoint_grid_rows_spec + array_row_indices_disjoint compositional slice).