{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/","md":"https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1","date":"2026-09-03","date_precision":"day","short_title":"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","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"GitHub: tadamcz/erdos1207 (Lean proof of Erdős #1207 by GPT-6 Astra)","url":"https://github.com/tadamcz/erdos1207","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"arXiv 2609.25050: FrontierMath Erdős (Adamczewski, Bloom)","url":"https://arxiv.org/abs/2609.25050","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"GitHub: tadamcz/erdos1 (Lean disproof)","url":"https://github.com/tadamcz/erdos1","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"GitHub: tadamcz/erdos571 (rational exponents, Lean proof)","url":"https://github.com/tadamcz/erdos571","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"arXiv 2610.18713: A proof of the rational exponents conjecture (Conlon, Janzer et al., expert write-up of #571)","url":"https://arxiv.org/abs/2610.18713","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"GitHub: tadamcz/erdos74","url":"https://github.com/tadamcz/erdos74","type":"code","group":"primary","domain":"github.com"},{"n":7,"title":"GitHub: tadamcz/erdos126","url":"https://github.com/tadamcz/erdos126","type":"code","group":"primary","domain":"github.com"},{"n":8,"title":"Quanta: Why Erdős Problems Are Falling to AI (3 Aug 2026)","url":"https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/","type":"press","group":"press","domain":"quantamagazine.org"},{"n":9,"title":"erdosproblems.com #1 (status: disproved, Lean)","url":"https://www.erdosproblems.com/1","type":"discussion","group":"reactions","domain":"erdosproblems.com"},{"n":10,"title":"Thomas Bloom on X: A big day for AI and mathematics (3 Sep 2026)","url":"https://x.com/thomasfbloom/status/2095630765035864260","type":"discussion","group":"reactions","domain":"x.com"}],"official":7,"filed":"2026-09-30","updated":"2026-10-16","orgs":["OpenAI","Epoch AI"],"title":"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","summary":"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"],"key_numbers":[],"tags":["math","erdos-problems","lean","counterexample","combinatorics","number-theory","astra","benchmark"],"science":{"field":"mathematics","subfield":"additive combinatorics / extremal graph theory","problem":"Erdős problems #1 (distinct subset sums), #74, #126 and #571 (rational exponents conjecture)","result":"#1 disproved (N ≤ ε·2^n possible for arbitrarily large n); #571 proved (every rational α in [1,2) is a Turán exponent of a bipartite graph); #74 answered in the negative direction; #126 proved with |S(A)| ≫ n^(1/2).","open_since":"1931","ai_system":["GPT-6 Astra (pre-release)"],"human_role":"Autonomous: the model worked in an agent harness without human steering; humans chose the problems, wrote the Lean statements and wrote informal expositions afterwards","verification":{"key":"lean","label":"Lean-verified","text":"Formal proofs in Lean 4 checked against audited challenge statements; informal human digestion ongoing; not peer-reviewed"},"status":"confirmed","shock":"The oldest open Erdős problem, which Erdős posed at 18, fell to an autonomous agent run as a side effect of building a benchmark.","short":"Erdős problems #1, #74, #126 and #571"},"body_md":"## What happened\n\nEpoch 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.\n\n## Why it matters\n\nProblem #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\".","disputed":[],"related":[{"id":"2026-10-13-rational-exponents-conjecture-proof-paper-conlon","url":"https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/","date":"2026-10-13","date_precision":"day","short_title":"Eight extremal combinatorialists write up GPT-6 Astra's proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.)","deck":null,"takeaway":"The rational exponents proof that GPT-6 Astra found during Epoch AI's FrontierMath Erdős runs now has a human-written paper by leading experts in the area, who explain and situate it and call it one of 'the recent AI-driven breakthroughs'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Caltech","EPFL","KAIST","IBS","Emory University","Epoch AI","OpenAI"]},{"id":"2026-09-21-openai-100-open-problems-claim","url":"https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/","date":"2026-09-21","date_precision":"day","short_title":"OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)","deck":null,"takeaway":"If substantiated, it would mean open problems are being resolved at industrial scale.","category":"science","category_label":"Science & math","importance":3,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-29","updated":"2026-10-10","orgs":["OpenAI"]},{"id":"2026-09-10-astra-leanopenproblems-september-results","url":"https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/","date":"2026-09-10","date_precision":"day","short_title":"GPT-6 Astra's Epoch AI run adds more Lean-checked results","deck":"Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved","takeaway":"Autonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-07-koethe-conjecture-disproved","url":"https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/","date":"2026-09-07","date_precision":"day","short_title":"Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample","deck":null,"takeaway":"If it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra proves the Erdős–Sós conjecture with a short counting argument","deck":"Mathematicians race to simplify and extend it","takeaway":"Erdős–Sós is one of the best-known conjectures in extremal graph theory, and this is among the clearest cases of an AI finding a genuinely new, short idea that experts call \"ingenious\" and \"surprising\".","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":9,"filed":"2026-09-30","updated":"2026-09-30","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-03-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"OpenAI releases GPT-6 Astra, its first GPT-6 model","deck":null,"takeaway":"Astra is the first GPT-6-generation model and the first frontier release after the Hugging Face sandbox-escape incident and OpenAI's August training pause.","category":"model-release","category_label":"Model releases","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":20,"official":11,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2025-12-08-ai-erdos-problems-wave-late-2025","url":"https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/","date":"2025-12-08","date_precision":"day","short_title":"Genuine AI-assisted solutions to Erdős problems begin","deck":"#124 (Aristotle), #1026 (48-hour human–AI collaboration)","takeaway":"Erdős problems became the first large, public, verifiable scoreboard for AI in research mathematics, and set the stage for 2026's much larger results.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Harmonic","Google DeepMind","OpenAI"]}],"people":[{"id":"thomas-bloom","name":"Thomas Bloom","url":"https://postcutoff.com/person/thomas-bloom/"}],"posts":[{"id":"2026-09-03-thomasfbloom-frontiermath-erdos","title":"Thomas Bloom: A big day for AI and mathematics — FrontierMath Erdős","url":"https://postcutoff.com/p/2026-09-03-thomasfbloom-frontiermath-erdos/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-16","type":"edit","text":"Added Erdős #1207 (repo tadamcz/erdos1207), found while checking the Oct 16 lead on Epoch's LeanOpenProblems runs"},{"date":"2026-10-14","type":"edit","text":"Added the expert write-up of the #571 proof (arXiv 2610.18713) as a follow-up"},{"date":"2026-10-07","type":"edit","text":"Science block: short label (normalised Erdős problem labels)"},{"date":"2026-09-30","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-30","run":null,"sources_read":null,"updated":"2026-10-16","human_review":null,"version":{"date":"2026-10-16"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":126,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":65,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":156,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":95,"in_training_data":false}],"short_url":"https://postcutoff.com/s/pre-release-gpt-6-astra"}