{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-10-30-number-of-c4-free-graphs-erdos-question-tianchi-yang-gpt-6-pro/","md":"https://postcutoff.com/e/2026-10-30-number-of-c4-free-graphs-erdos-question-tianchi-yang-gpt-6-pro/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":"2026-10-30-number-of-c4-free-graphs-erdos-question-tianchi-yang-gpt-6-pro","date":"2026-10-30","date_precision":"day","short_title":"Number of C₄-free graphs is 2^{(1+o(1))ex(n,C₄)} (claim, GPT-6 Pro)","deck":null,"takeaway":"Tianchi Yang's preprint (arXiv 2610.39595, 29 Oct 2026) claims that the number of graphs on n vertices with no 4-cycle is 2^{(1+o(1))ex(n,C₄)}. That answers a question of Erdős. The best earlier upper bound was Kleitman and Winston's 2^{(1.0819…+o(1))n^{3/2}} from 1982, and Morris and Saxton showed the same statement fails for C₆. The AI statement says the author supplied the main ideas, a modified hypergraph container method, and 'GPT-6 Pro assisted in completing the proofs.' Single author, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.39595: The number of C₄-free graphs (Tianchi Yang)","url":"https://arxiv.org/abs/2610.39595","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"PDF with the AI statement","url":"https://arxiv.org/pdf/2610.39595","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic"],"title":"Counting C₄-free graphs: Tianchi Yang claims 2^{(1+o(1))ex(n,C₄)}, answering a question of Erdős; GPT-6 Pro 'assisted in completing the proofs'","summary":"arXiv 2610.39595 (\"The number of C₄-free graphs\", Tianchi Yang) claims that there are 2^{(1+o(1))ex(n,C₄)} C₄-free graphs on n vertices, answering a question of Erdős. Erdős–Kleitman–Rothschild and Erdős–Frankl–Rödl proved the analogue for non-bipartite H, Kleitman–Winston gave 2^{O(n^{3/2})} for C₄, and Morris–Saxton showed the analogue fails for C₆. The proof uses a modified hypergraph container method and a new bound on the number of matchings. The author gave the main ideas and GPT-6 Pro helped complete the proofs.","key_facts":["Claim (abstract): 'Erdős, Frankl and Rödl showed that the number of graphs on n vertices with no copy of a non-bipartite graph H is 2^{(1+o(1))ex(n,H)}, whereas Morris and Saxton showed that this fails for C₆. Answering a question of Erdős, we prove that the number of C₄-free graphs on n vertices is 2^{(1+o(1))ex(n,C₄)}.'","AI statement (verbatim): 'The author provided the main ideas of the proof, combining the hypergraph container method with the deferral of diagonal constraints. GPT-6 Pro assisted in completing the proofs, and GPT 6.1 and Opus 5.5 were used to revise the manuscript and improve its exposition. The author has verified all mathematical arguments and results and takes full responsibility for the content of this paper.'","Earlier bounds (paper, §1): Kleitman and Winston proved at most 2^{(1.0819…+o(1))n^{3/2}} C₄-free graphs; Balogh–Samotij and Morris–Saxton gave 2^{O(ex(n,H))}-type bounds for K_{s,t} and C_{2ℓ}; Morris–Saxton found 2^{(1+c)ex(n,C₆)} C₆-free graphs for infinitely many n, so the (1+o(1)) form is not automatic for bipartite H","Method: a hypergraph container argument that treats pairs of vertices with large codegree separately, plus a new bound on the number of matchings in a graph","Author: Tianchi Yang (Georgia Tech), who earlier disproved the Erdős–Sós bipartite-link conjecture with GPT-6.1 (arXiv 2610.11642; see the October results entry)","Verification status: single-author arXiv preprint, not refereed, no formal proof; no HN thread or specialist comment found on 31 Oct"],"key_numbers":[],"tags":["math","combinatorics","extremal-graph-theory","erdos-problems","hypergraph-containers","gpt-6-pro","gpt-6-1","claude-opus-5-5","preprint"],"science":{"field":"mathematics","subfield":"extremal and probabilistic combinatorics","problem":"Erdős's question: is the number of C₄-free graphs on n vertices 2^{(1+o(1))ex(n,C₄)}?","result":"Claimed affirmative answer (previous best upper bound 2^{1.0819…n^{3/2}}, Kleitman–Winston 1982)","open_since":"1982","ai_system":["GPT-6 Pro","GPT-6.1","Claude Opus 5.5"],"human_role":"Human-led: main ideas by the author; GPT-6 Pro helped complete the proofs; GPT-6.1 and Opus 5.5 revised the text","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed single-author preprint"},"status":"pending","shock":"A counting question that resisted the container method for C₄ is claimed in one short paper with a model completing the proofs."},"body_md":"## What happened\n\nHow many graphs on n vertices contain no 4-cycle? There are at least 2^{ex(n,C₄)}, the subgraphs of one extremal graph. Kleitman and\nWinston showed in 1982 that the total is at most 2^{1.0819…n^{3/2}}, the same order. Erdős asked whether the exponent is asymptotically\nexactly ex(n,C₄). For C₆ the analogous statement is false (Morris–Saxton).\n\nOn 29 Oct Tianchi Yang posted a proof that the answer is yes. The author gives the main ideas and says GPT-6 Pro helped complete the\nproofs.\n\n## Why it matters\n\nIt settles, if correct, a well-known counting question in extremal graph theory, with the AI role stated by the author. It is a\npreprint.","disputed":[{"label":"Verification","text":"Unrefereed single-author preprint","sources":[]}],"related":[{"id":"2026-10-31-ai-assisted-results-october-31-listing","url":"https://postcutoff.com/e/2026-10-31-ai-assisted-results-october-31-listing/","date":"2026-10-31","date_precision":"day","short_title":"AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct","deck":null,"takeaway":"The 31 Oct sweep listed 344 arXiv papers that mention AI, 75 conjecture papers with no AI statement and 48 uncited Hexagon works. We skimmed every AI statement and read about 70 PDFs or Hexagon records. Seven results got their own entries. Two come from senior authors: Panchenko's Mézard–Parisi formula ('found by ChatGPT Astra') and Bhargava's positive-density cyclicity of elliptic curves, written with GPT-6. The other five are Tianchi Yang's count of C₄-free graphs, Kolpakov's sharp dimension bounds for right-angled hyperbolic polytopes, the generalized Lax conjecture (Kummer–Netzer's AI-found generic case and OpenAI's counterexample), Diening–Storn's AFEM optimality for every bulk parameter, and an unverified GPT-6 Astra claim on Onsager's hard rods. The rest are listed here, among them Pongrácz's R(5,6) ≤ 84 certificate with GPT-6 Astra and Bowler–Pitz's negative answers to Halin-type questions, where Claude found key lemmas.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":44,"official":42,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-10-13-ai-assisted-results-october-2026","url":"https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/","date":"2026-10-13","date_precision":"day","short_title":"October 2026: the arXiv stream of conjectures settled with disclosed AI help continues","deck":null,"takeaway":"In the first nine days of October 2026 the stream did not slow down: each arXiv listing brought several named conjectures with a disclosed AI role, and two new patterns appeared.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":17,"official":17,"filed":"2026-10-13","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-31","type":"filed","text":"Created (science agent; arXiv abstract, introduction and AI statement read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-31","run":null,"sources_read":"Science agent; arXiv abstract, introduction and AI statement read","updated":"2026-10-31","human_review":null,"version":{"date":"2026-10-31"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":183,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":122,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":213,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":152,"in_training_data":false}],"short_url":"https://postcutoff.com/s/number-c4-free-graphs-2"}