{"schema":"postcutoff/event@1","as_of":"2026-10-14T22:23:00+02:00","url":"https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/","md":"https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/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":"2026-10-13-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":[{"n":1,"title":"arXiv 2610.18713: A proof of the rational exponents conjecture (Conlon, Janzer, Jiang, Kim, Lee, Liu, Longbrake, Yepremyan)","url":"https://arxiv.org/abs/2610.18713","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GitHub: tadamcz/erdos571 (Lean proof from the Epoch AI run)","url":"https://github.com/tadamcz/erdos571","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"erdosproblems.com: Problem #571","url":"https://www.erdosproblems.com/571","type":"discussion","group":"reactions","domain":"erdosproblems.com"}],"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Caltech","EPFL","KAIST","IBS","Emory University","Epoch AI","OpenAI"],"title":"Eight extremal combinatorialists write up GPT-6 Astra's proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.)","summary":"arXiv 2610.18713 (13 Oct 2026), \"A proof of the rational exponents conjecture\", by David Conlon, Oliver Janzer, Tao Jiang, Jaehoon Kim, Joonkyung Lee, H. Liu, Sean Longbrake and Liana Yepremyan, presents and explains the proof of the Erdős–Simonovits rational exponents conjecture (Erdős problem #571) that Epoch AI reported GPT-6 Astra found during FrontierMath Erdős benchmarking in September. The authors, who did much of the earlier human work on the problem, write that \"this advance draws heavily on ideas and methods developed by many researchers over several years\". It is the first human-written paper on the result; the original came as a Lean proof and an informal write-up.","key_facts":["Paper: arXiv 2610.18713 [math.CO], 13 Oct 2026, 13 pages; authors from Caltech (Conlon), EPFL (Janzer), Miami University (Jiang), KAIST (Kim), KIAS (Lee), IBS Daejeon (Liu) and Emory (Longbrake, Yepremyan)","Theorem 1.2: for every rational r in [1, 2] there is a graph H with ex(n, H) = Θ(n^r), the Erdős–Simonovits rational exponents conjecture","Framing: 'we wish to explain and situate a proof of the conjecture for all exponents claimed by Epoch AI [1], obtained during benchmarking for the recently released model GPT-6 Astra.' Reference [1] is Thomas Bloom's erdosproblems.com/571 page","The authors name four sources the proof builds on: the Bukh–Conlon lower bound, an Erdős–Simonovits result on powers of rooted graphs, the Kang–Kim–Liu subdivision conjecture, and a technique of Conlon, Janzer and Lee","Key technical result (Theorem 1.3): a variant of the Kang–Kim–Liu subdivision conjecture for the graphs H_k (each edge replaced by a path of length k plus two apex vertices), which lets one 'percolate' from the exponent 1 to all rational exponents","Before the AI proof, the conjecture was known for exponents 1 + a/b with b > a² and 2 − a/b with b ≥ (a − 1)², among other ranges","Still open, per the concluding remarks: the Bukh–Conlon conjecture on rooted powers of trees and the Kang–Kim–Liu subdivision conjecture, both of which would imply the result","AI use in this paper: 'We used ChatGPT Pro to draw the figures in this paper and to proofread an early draft.'"],"key_numbers":[],"tags":["math","combinatorics","extremal-graph-theory","turan-numbers","erdos-problems","gpt-6-astra","epoch-ai","exposition"],"science":{"field":"mathematics","subfield":"extremal graph theory","problem":"Erdős–Simonovits rational exponents conjecture (Erdős problem #571): every rational r in [1, 2] is the Turán exponent of some graph","result":"Human-written exposition of the AI-found proof: for every rational r in [1, 2] there is a graph H with ex(n, H) = Θ(n^r).","open_since":"","ai_system":["GPT-6 Astra"],"human_role":"The proof was found by GPT-6 Astra in Epoch AI's FrontierMath Erdős runs (Sept 2026); eight human experts wrote, explained and situated it in this paper","verification":{"key":"lean","label":"Lean-verified","text":"Lean proof from the Epoch AI run (tadamcz/erdos571) plus an expert-written preprint; not yet refereed"},"status":"confirmed","shock":"A central conjecture of extremal graph theory, attacked for over a decade by the authors' own programme, was closed by a model during a benchmark run."},"body_md":"## What happened\n\nIn September, Epoch AI and Thomas Bloom reported that GPT-6 Astra, in larger-budget runs outside the FrontierMath Erdős benchmark, had proved the Erdős–Simonovits rational exponents conjecture (#571) with a Lean proof (see [the FrontierMath Erdős entry](/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/)). The benchmark report called the informal write-ups placeholders until experts prepared proper papers.\n\nOn 13 Oct 2026 eight extremal combinatorialists posted that paper, [arXiv 2610.18713](https://arxiv.org/abs/2610.18713). Several of them wrote the earlier partial results the proof builds on, including the Bukh–Conlon lower bound, the Kang–Kim–Liu subdivision conjecture and the Conlon–Janzer–Lee technique. The abstract says: \"We present a proof of this conjecture claimed by Epoch AI, explaining how it builds upon ideas and methods developed by many researchers.\" The introduction adds: \"As with many of the recent AI-driven breakthroughs, this advance draws heavily on ideas and methods developed by many researchers over several years.\"\n\nThe core is Theorem 1.3, a variant of the Kang–Kim–Liu subdivision conjecture for graphs H_k (each edge of H replaced by a path of length k, plus two vertices joined to the two sides of the bipartition). Combined with the Erdős–Simonovits result on powers of balanced rooted graphs and the Bukh–Conlon lower bound, it reaches every rational exponent starting from the exponent 1. The two stronger conjectures (Bukh–Conlon on rooted tree powers, Kang–Kim–Liu on 1-subdivisions) remain open.\n\nThe paper's own AI statement covers only figures and proofreading by ChatGPT Pro. No Hacker News thread was found on 14 Oct (Algolia search); X was not searched directly.\n\n## Why it matters\n\nIt is the second wave of one of the strongest AI-found results of 2026. Experts in the area accept the proof, rewrite it for humans and place it in the literature, while making the point that the model built on their programme. Its verification moves from a Lean certificate plus placeholder write-up to an expert-written preprint.","disputed":[],"related":[{"id":"2026-10-14-ai-assisted-results-mid-october-2026","url":"https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/","date":"2026-10-14","date_precision":"day","short_title":"Mid-October 2026: arXiv results with disclosed AI help","deck":null,"takeaway":"The 13 Oct arXiv listing alone brought a human write-up of GPT-6 Astra's rational exponents proof, Ulam's Lie group question answered with an Astra construction, a better-than-2 k-median algorithm, and about a dozen smaller results whose AI statements range from 'suggested by ChatGPT' to 'GPT 6 Pro was used … to obtain all other new results'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":12,"filed":"2026-10-14","updated":"2026-10-14","orgs":["OpenAI","Anthropic","various mathematicians"]},{"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/","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":"Problem #1 is probably the longest-standing open Erdős problem.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":6,"filed":"2026-09-30","updated":"2026-10-14","orgs":["OpenAI","Epoch AI"]}],"people":[{"id":"thomas-bloom","name":"Thomas Bloom","url":"https://postcutoff.com/person/thomas-bloom/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-14","type":"filed","text":"Created from the arXiv paper (arXiv AI-disclosure list)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-14","run":null,"sources_read":"The arXiv paper (arXiv AI-disclosure list)","updated":"2026-10-14","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":166,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":105,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":196,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":135,"in_training_data":false}],"short_url":"https://postcutoff.com/s/eight-extremal-combinatorialists"}