Erdős Prize Problems — The Open Bounties

All Open Prize Erdős Problems

58 open problems • $37,645 total • Ready for AI

Prize	#	Problem	Problem Comments	Solver	Gemini-3-Pro Deepthink	GPT-5.2 Pro Extended thinking	Synth Data	Problem in Lean	Evaluator	Eval of Deepthink by GPT-5.2	Eval of GPT-5.2 by Deepthink	Proof in Lean
$10,000	142	Asymptotic formula for r_k(N) - arithmetic progression-free sets	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$5,000	3	Arithmetic progressions in sets with divergent reciprocal sum	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$1,000	20	Sunflower Conjecture - k-sunflowers in uniform set families	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$1,000	30	Sidon Sets - maximum size h(N) bounds	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$1,000	64	Power-of-two cycles in graphs with min degree ≥3	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$1,000	592	Partition ordinals - Ramsey properties of countable ordinals	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$1,000	625	Cochromatic number - χ(G) - ζ(G) growth for random graphs	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$1,000	687	Jacobsthal function Y(x) bounds	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	1	Subset sums distinctness - N ≫ 2^n conjecture	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	28	Sumset density - if A+A cofinite then limsup 1_A∗1_A(n)=∞	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	39	Infinite Sidon set growth rate N^{1/2-ε}	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	40	Generalized sumset density with growth function g(N)	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	41	Triple sum distinctness - liminf \|A∩{1,...,N}\|/N^{1/3}=0?	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	66	Convolution limit - does lim 1_A∗1_A(n)/log n exist and ≠0?	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	74	Infinite chromatic graphs with locally bipartite subgraphs	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	89	Distinct distances - ≫ n/√(log n) distances? AI EVOLVING	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	90	Unit distance problem - max pairs at distance 1 AI EVOLVING	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	92	Equidistant points - is f(n) ≤ n^{o(1)}?	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	107	Happy ending problem - f(n)=2^{n-2}+1 for convex polygons?	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	138	van der Waerden numbers - prove W(k)^{1/k}→∞	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	143	Sparse sets with \|kx-y\|≥1 constraint	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	146	Bipartite r-degenerate Turán exponent 2-1/r	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	161	Ramsey discontinuity in hypergraph colorings	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	500	3-uniform hypergraph Turán number ex_3(n,K_4^3)	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	564	3-uniform hypergraph Ramsey R_3(n) ≥ 2^{2^{cn}}?	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	593	High chromatic 3-uniform hypergraph characterization	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	601	Ordinal Ramsey - infinite paths or independent sets	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	604	Pinned distance problem - ≫ n^{1-o(1)} distances?	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	712	Hypergraph Turán density ex_r(n,K_k^r)/C(n,r)	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	713	Bipartite Turán exponent rationality	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$500	1135	Collatz Conjecture - f^{(k)}(m)=1 for all m?	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	18	Practical numbers and divisor sums h(m) bounds	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	50	Density of integers where φ(n)<cn is purely singular	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	52	Sum-product problem - max(\|A+A\|, \|AA\|) bounds	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	77	Find lim R(k)^{1/k} for Ramsey numbers	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	114	Polynomial with max curve length when p(z)=z^n-1	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	123	d-completeness of {a^k b^l c^m} for coprime a,b,c	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	126	Distinct prime factors in products of pairwise sums	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	165	Asymptotic formula for R(3,k)	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	183	Multicolor Ramsey limit R(3;k)^{1/k}	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	595	Infinite K₄-free graph not union of countable triangle-free	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$250	671	Lagrange interpolation convergence despite unbounded constants	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	43	Two Sidon sets with disjoint difference sets	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	78	Constructive proof that R(k) > C^k	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	86	C₄-free subgraphs of hypercube Q_n	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	97	Convex polygon vertex equidistance property	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	99	Unit distance sets containing equilateral triangles	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	101	Lines through exactly four points among n points	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	104	Unit circles through three or more points	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	119	Polynomial maximum on unit circle	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	120	Measure avoiding scaled copies of infinite sets	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	132	Multiple distances in point configurations	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	241	Maximum sets with distinct triple sums	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	552	Ramsey number R(C₄, S_n) bounds	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	588	Lines through k points in point configurations	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	708	Products and divisibility in integer intervals	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$100	1029	Ramsey number growth rate R(k)/(k·2^{k/2})	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$78	710	Minimal f(n) for divisibility k\|a_k in interval	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$78	711	Related divisibility function f(n,m)	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$50	32	Additive complement to primes with minimal size	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$50	661	Distinct distances between two sets minimization	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$44	647	Robin's inequality type - τ(n) divisor count	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$25	7	Distinct odd covering system existence	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$25	633	Triangles into square number of congruent pieces	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$25	634	Find all n for triangle into n congruent triangles	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$10	470	Existence of odd weird numbers	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
$10	1052	Finiteness of unitary perfect numbers	Problem Comments Explainer		NA	NA	NA	NA		NA	NA	NA
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AI progress

18+ AI solutions • 30+ partial • 4 under evolution • 40+ formalizations

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$500 ERD-90 AI EVOLVING

Unit Distance Problem

Maximize the number of unit distances among n points in the plane. Related to incidence geometry and the Szemerédi-Trotter theorem.

u(n) = Ω(n^{1+c/log log n}) to O(n^{4/3})

ShinkaEvolve Progress 121 → 125+ unit distances (n=50)

Geometry Combinatorics 🧬 Evolving

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$500 ERD-89 AI EVOLVING

Distinct Distances Problem

Minimize the number of distinct pairwise distances among n points. Solved theoretically by Guth-Katz (2015), optimal constructions remain open.

g(n) ≥ cn/√(log n) • Integer lattice achieves ~n/√(log n)

ShinkaEvolve Progress Searching for optimal configurations

Geometry Guth-Katz 🧬 Evolving

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Open HEILBRONN AI EVOLVING

Heilbronn Triangle Problem

Place n points in a unit square to maximize the minimum triangle area. Bounds: Ω(log n / n²) to O(1/n^{8/7-ε}).

Δ(n) = max min_{i,j,k} Area(p_i, p_j, p_k)

ShinkaEvolve Progress Optimizing point configurations

Geometry Optimization 🧬 Evolving

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$250 SUM-FREE AI EVOLVING

Sum-Free Sets Problem

Find the largest sum-free subset of {1, 2, ..., n}. A set is sum-free if no two elements sum to another element.

f(n) ≈ n/3 • No a + b = c for a, b, c ∈ S

ShinkaEvolve Progress Score: 42.6 (n=100)

Number Theory Combinatorics 🧬 Evolving

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$5,000 ERD-AP OPEN

Arithmetic Progressions Conjecture

If the sum of reciprocals of a set of positive integers diverges, must the set contain arbitrarily long arithmetic progressions?

If Σ(1/a) = ∞ for A ⊂ ℤ⁺, then ∀k ∃ a,d: {a, a+d, a+2d, ..., a+(k-1)d} ⊂ A

Progress Partial Results

Number Theory Combinatorics

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$1,000 ERD-T1C PARTIAL

Sunflower Conjecture

Among C^n sets with n elements, must three exist whose mutual intersection equals each pairwise intersection?

∃C: |F| > Cⁿ, |S|=n ∀S∈F ⟹ ∃S₁,S₂,S₃∈F: S₁∩S₂ = S₂∩S₃ = S₁∩S₃ = S₁∩S₂∩S₃

Progress Major Breakthrough 2019

Set Theory Combinatorics

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$500 ERD-P2 OPEN

Distinct Subset Sums

In sets where all subsets have distinct sums, what is the minimum possible value of the largest element?

max(A) ≥ c·2ⁿ for some c > 0, when all 2ⁿ subset sums are distinct

Progress c ≈ 0.22 proven

Number Theory Optimization

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$500 ERD-T9N SOLVED

Discrepancy Coloring

In any 2-coloring of positive integers, can the imbalance of colors among multiples of any number remain bounded?

χ: ℤ⁺→{-1,+1}, sup_d |Σₖ≤n χ(kd)| = O(1) possible?

Progress Disproved 2015

Combinatorics Discrepancy

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$500 ERD-T7N OPEN

Asymptotic Bases

For an asymptotic basis of order 2, can the number of representations of integers as sums of two elements remain bounded?

A+A ⊇ {n : n > N}, sup_n |{(a,b)∈A²: a+b=n}| < ∞ possible?

Progress Open

Additive Combinatorics Number Theory

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Full Solutions by AI

SOLVED ERD-205

Problem 205

Complete Lean formalization achieved.

Aristotle

ChatGPT 5.2

📅 Jan 10, 2026

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SOLVED ERD-281

Problem 281

Complete via locating Davenport & Erdős (1936) + Rogers theorem.

GPT 5.2 Pro

📅 Jan 17, 2026

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SOLVED ERD-333

Problem 333

Full Lean proof matching Erdős & Newman (1977).

ChatGPT 5.2

Claude Opus 4.5

📅 Dec 25, 2025

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SOLVED ERD-347

Problem 347

Human-AI collaboration with Terence Tao, van Doorn, Naskrecki.

GPT Codex

Aristotle

📅 Oct 2025–Jan 2026

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SOLVED ERD-397

Problem 397

Complete Lean formalization; related to Elkies (2013).

Aristotle

ChatGPT 5.2

📅 Jan 10, 2026

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SOLVED ERD-401

Problem 401

Collaboration: Alexeev, Barreto, Leeham, Sothanaphan.

Aristotle

ChatGPT 5.2

📅 Jan 10-11, 2026

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SOLVED ERD-652

Problem 652

Complete by applying existing theorem.

GPT 5.2 Pro

📅 Jan 17, 2026

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SOLVED ERD-659

Problem 659

Full Lean solution by Benjamin Grayzel.

Gemini 3.0

📅 Jan 13, 2026

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SOLVED ERD-728

Problem 728

Complete proof; surpassed Pomerance (2014) partial.

Aristotle

ChatGPT 5.2

📅 Jan 6, 2026

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SOLVED ERD-729

Problem 729

Full Lean solution building on problem 728.

Aristotle

ChatGPT 5.2

📅 Jan 8-10, 2026

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SOLVED ERD-848

Problem 848

By Mehtaab Sawhney & Mark Sellke.

GPT-5

📅 Oct-Nov 2025

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SOLVED ERD-871

Problem 871

Upgraded Erdős & Nathanson (1989) to complete Lean proof.

Claude Opus 4.5

Gemini 3 Pro

📅 Jan 5, 2026

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SOLVED ERD-897

Problem 897

Full Lean proof matching Wirsing (1981).

Aristotle

Archivara

📅 Dec 26, 2025

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SOLVED ERD-958

Problem 958

New Lean proof; matched Clemen, Dumitrescu, Liu (2025).

Aristotle

📅 Dec 27, 2025

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SOLVED ERD-1007

Problem 1007

Complete Lean proof; related to House (2013).

Aristotle

📅 Jan 19, 2026

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SOLVED ERD-1026

Problem 1026

Collaboration: Alexeev, Cambie, Tao, Wu. Matched Tidor, Wang, Yang (2016).

ChatGPT

Aristotle

AlphaEvolve

📅 Dec 8, 2025

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SOLVED ERD-1043

Problem 1043

Complete Lean proof; matched Pommerenke (1961).

Aristotle

📅 Dec 28, 2025

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SOLVED ERD-1047

Problem 1047

New Lean proof of Goodman (1966) result.

Aristotle

📅 Jan 21, 2026

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Partial Solutions & Progress

PARTIAL ERD-42

Problem 42

Partial Lean result with computational verification.

GPT-5.2 Pro Extended thinking

Grok

📅 Jan 19-20, 2026

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PARTIAL ERD-43

Problem 43

New Lean proof of Barreto (2025) partial result.

Aristotle

📅 Dec 4, 2025

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PARTIAL ERD-106

Problem 106

Matched Halász (1984) construction.

AlphaEvolve

📅 Nov 3, 2025

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PARTIAL ERD-124

Problem 124

Partial Lean result achieved.

Aristotle

📅 Nov 29, 2025

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PARTIAL ERD-198

Problem 198

New proof of Baumgartner (1975).

AlphaProof

📅 2025

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PARTIAL ERD-367

Problem 367

Alexeev, van Doorn, Tao collaboration.

Gemini DeepThink

Aristotle

📅 Nov 20-22, 2025

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PARTIAL ERD-477

Problem 477

Lean proof of variant problem.

AlphaProof

📅 Jan 7, 2026

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PARTIAL ERD-507

Problem 507

Surpassed Friedman (2015) constructions.

AlphaEvolve

📅 Nov 3, 2025

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IN PROGRESS ERD-1038

Problem 1038

Koizumi, jspier, Sothanaphan, Tao collaboration - ongoing.

ChatGPT

AlphaEvolve

📅 In Progress

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PARTIAL ERD-1097

Problem 1097

Slight improvement to Lemm (2015) construction.

AlphaEvolve

📅 Nov 3, 2025

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+ 20 more partial results including problems 224, 264, 379, 460, 486, 488, 493, 514, 524, 563, 679, 684, 788, 850, 868, 1077, 1095...

Failed Attempts (Incorrect Proofs)

FAILED ERD-51

Problem 51

Incorrect proof submitted.

ChatGPT (free)

📅 Jan 11, 2026

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FAILED ERD-233

Problem 233

Incorrect conditional proof.

ChatGPT 5.2 Pro

📅 Jan 18, 2026

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FAILED ERD-616

Problem 616

Incorrect proof from multiple AI systems.

ChatGPT 5.2

Gemini 3 Pro

Claude 4.5

📅 Jan 18, 2026

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FAILED ERD-888

Problem 888

Incorrect proof from multiple frontier models.

GPT 5.2

Opus 4.5

Gemini 3 Pro

📅 Jan 18, 2026

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FAILED ERD-1044

Problem 1044

Incorrect claim made.

ChatGPT 5.2 Pro

📅 Jan 5, 2026

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Note: Failed attempts are valuable for understanding AI limitations and improving future systems.

📐 AI Formalizations in Lean

40+ existing mathematical proofs formalized into Lean code by AI systems.

[26] Ruzsa proof

[31] Lorentz (1954)

[56] Ahlswede-Khachatrian

[94] Lefmann & Thiele

[105] Xichuan (2025)

[189] Kovač (2023)

[226] Sato & Rankin

[229] Barth & Schneider

[246] Birch (1959)

[275] Balister et al.

[290] van Doorn (2024)

[303] Brown & Rödl

[337] Ruzsa & Turjányi

[350] Ryavec (1974)

[370] Steinerberger

[418] Browkin & Schinzel

[480] Chung & Graham

[481] Barreto (2025)

[499] Marcus & Minc

[541] Grynkiewicz

[613] Pikhurko (2001)

[645] Brown & Landman

[678] Cambie (2025)

[698] Bergman (2011)

[707] Hall (1947)

[845] van Doorn & Everts

[862] Saxton & Thomason

[967] Yip (2025)

[1000] Haight

[1008] Conlon, Fox, Sudakov

[1028] Erdős (1963)

[1034] Ma-Tang (2025)

[1036] Shelah (1998)

[1037] Cambie, Chan, Hunter

[1080] De Caen & Székely

Primary AI systems for formalization: Aristotle, ChatGPT, SeedProver 1.5, Bytedance Seed AI4Math