---
id: "2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1"
url: "https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-09-03"
date_precision: day
category: science
importance: 5
confidence: high
status: [Result confirmed]
verification: Lean-verified
sources: 10
editor: human editor
human_review: null
version: "2026-10-16"
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/

# Pre-release GPT-6 Astra disproves Erdős's 'first serious problem' (1931, $500) and proves the rational-exponents conjecture, all Lean-verified, in Epoch's FrontierMath Erdős runs

In Epoch AI's FrontierMath Erdős runs (announced 3 Sep 2026), a pre-release GPT-6 Astra autonomously resolved five of 68 hand-picked open Erdős problems with Lean-checked proofs. They include a disproof of Erdős problem #1 on distinct subset sums, which Erdős dated to 1931 and called 'perhaps my first serious problem', and a proof of the Erdős–Simonovits rational-exponents conjecture for bipartite Turán numbers (#571). The Erdős–Sós conjecture (#548) is in a separate entry.

## Key facts
- Benchmark: 68 open Erdős problems chosen by Thomas Bloom from 652 open problems on erdosproblems.com; a resolution counts only if it is proved in Lean; default budget $300 and 72 hours per problem
- Official score (one attempt each): pre-release GPT-6 Astra 3% (#74 disproved for $218 in 15 h; #126 proved for $247 in 16 h); GPT-5.6 Sol, GPT-5.5, Claude Fable 5.1 and Claude Fable 5 all 0%
- Extra, non-benchmark attempts with larger budgets resolved 5 problems in total (#1, #74, #126, #548, #571), using over $220,000 of compute versus about $20,000 for the benchmark run
- #1 (Erdős, dated 1931, $500 prize): if A ⊆ {1..N} has n elements with all subset sums distinct, must N ≫ 2^n? Astra showed that for every ε > 0 there are arbitrarily large n with N ≤ ε·2^n, so the conjecture is false. The previous best construction was N ≤ 0.22002·2^n (Bohman). The proof is ineffective (no explicit n for a given ε). Found in 2 of 4 attempts ($405 / 27 h and $1,384 / 84 h)
- #571 (Erdős–Simonovits): for every rational α in [1,2) there is a bipartite graph G with ex(n; G) ≍ n^α. Bloom calls it 'the most difficult of the five solutions'. Found in 1 of 3 attempts ($617, 41 h)
- #74 (Erdős–Hajnal–Szemerédi): disproved in the unexpected direction. If every n-vertex subgraph can be made bipartite by deleting at most f(n) edges with f growing slowly enough, the graph has chromatic number ≤ 3
- #126 (Erdős–Turán): |S(A)| ≫ n^(1/2) primes divide the pairwise sums of an n-element set, improving the classical log n bound; Astra gave three distinct proofs (exponents 1/8, 1/3 and 1/2)
- erdosproblems.com now marks #1 as 'DISPROVED (LEAN)'
- The Lean repositories (tadamcz/erdos1, erdos74, erdos126, erdos571), created 3 Sep 2026, say the proofs were found autonomously in the benchmark harness; the report calls the informal write-ups 'placeholders' until human experts prepare proper papers
- Also published (repo tadamcz/erdos1207, created 7 Sept 2026): Erdős problem #1207 (isosceles-free subsets, attributed to Riddell) proved: there is c > 0 with P_2(n) < n^(1−c), i.e. for large n some n-point planar sets have no isosceles-free subset larger than n^(1−c) (a power saving that Erdős asked for); 'found autonomously by a pre-release version of GPT-6 Astra … running in the harness of the FrontierMath Erdős benchmark'; Lean 4 proof with an audit script; repository 'machine-written by AI assistants at the direction of Tom Adamczewski'; not counted in the five problems above and not independently reviewed

## What happened

Epoch AI and Thomas Bloom (erdosproblems.com) built FrontierMath Erdős, a benchmark of 68 open Erdős problems formalised in Lean, and ran five frontier models on it. Under the fixed $300 budget only the pre-release GPT-6 Astra solved anything (2 of 68). In further runs with larger budgets, which the authors report separately and do not count as benchmark results, Astra also disproved problem #1, proved the Erdős–Sós conjecture (#548) and proved the rational-exponents conjecture (#571). Every resolution comes with a Lean proof checked against a small challenge statement.

## Why it matters

Problem #1 is probably the longest-standing open Erdős problem. The #571 result settles a central question on Turán numbers of bipartite graphs. The paper also gives a denominator that most AI-math announcements lack. The fixed-budget run solved only 2 of 68 problems. Of the 63 unsolved problems, 56 were retried two to five times each (172 attempts) with no success, and the five resolutions took over $220,000 of compute. That makes it a sober counterweight to OpenAI's later claim of "100+ open problems".

## Follow-up: expert write-up of #571 (13 Oct)

On 13 Oct 2026 eight extremal combinatorialists (Conlon, Janzer, Jiang, Kim, Lee, Liu, Longbrake, Yepremyan) posted [arXiv 2610.18713](https://arxiv.org/abs/2610.18713), which explains and situates the #571 proof "claimed by Epoch AI, obtained during benchmarking for the recently released model GPT-6 Astra". It replaces the placeholder write-up with a human paper; see [its own entry](/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/).

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 126 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 65 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 156 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 95 days after its cutoff

## Sources

1. [GitHub: tadamcz/erdos1207 (Lean proof of Erdős #1207 by GPT-6 Astra)](https://github.com/tadamcz/erdos1207) (github.com, code)
2. [arXiv 2609.25050: FrontierMath Erdős (Adamczewski, Bloom)](https://arxiv.org/abs/2609.25050) (arxiv.org, paper)
3. [GitHub: tadamcz/erdos1 (Lean disproof)](https://github.com/tadamcz/erdos1) (github.com, code)
4. [GitHub: tadamcz/erdos571 (rational exponents, Lean proof)](https://github.com/tadamcz/erdos571) (github.com, code)
5. [arXiv 2610.18713: A proof of the rational exponents conjecture (Conlon, Janzer et al., expert write-up of #571)](https://arxiv.org/abs/2610.18713) (arxiv.org, paper)
6. [GitHub: tadamcz/erdos74](https://github.com/tadamcz/erdos74) (github.com, code)
7. [GitHub: tadamcz/erdos126](https://github.com/tadamcz/erdos126) (github.com, code)
8. [Quanta: Why Erdős Problems Are Falling to AI (3 Aug 2026)](https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/) (quantamagazine.org, press)
9. [erdosproblems.com #1 (status: disproved, Lean)](https://www.erdosproblems.com/1) (erdosproblems.com, discussion)
10. [Thomas Bloom on X: A big day for AI and mathematics (3 Sep 2026)](https://x.com/thomasfbloom/status/2095630765035864260) (x.com, discussion)

## Changes

- 2026-10-16: Added Erdős #1207 (repo tadamcz/erdos1207), found while checking the Oct 16 lead on Epoch's LeanOpenProblems runs
- 2026-10-14: Added the expert write-up of the #571 proof (arXiv 2610.18713) as a follow-up
- 2026-10-07: Science block: short label (normalised Erdős problem labels)
- 2026-09-30 (filed): Created

## Related

- 2026-10-13: [Eight extremal combinatorialists write up GPT-6 Astra's proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.)](https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/index.md)
- 2026-09-21: [OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)](https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/index.md)
- 2026-09-10: [GPT-6 Astra's Epoch AI run adds more Lean-checked results](https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/index.md)
- 2026-09-07: [Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample](https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/index.md)
- 2026-09-03: [GPT-6 Astra proves the Erdős–Sós conjecture with a short counting argument](https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/index.md)
- 2026-09-03: [OpenAI releases GPT-6 Astra, its first GPT-6 model](https://postcutoff.com/e/2026-09-03-gpt-6-astra/index.md)
- 2025-12-08: [Genuine AI-assisted solutions to Erdős problems begin](https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/index.md)
- People: [Thomas Bloom](https://postcutoff.com/person/thomas-bloom/)
- Archived: 1 post, listed in https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/index.json