{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-05-quasiplanar-graphs-linear-edges-chatgpt-6-pach-tardos-zeng/","md":"https://postcutoff.com/e/2026-11-05-quasiplanar-graphs-linear-edges-chatgpt-6-pach-tardos-zeng/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":"2026-11-05-quasiplanar-graphs-linear-edges-chatgpt-6-pach-tardos-zeng","date":"2026-11-05","date_precision":"day","short_title":"Quasiplanar graph conjecture proved by ChatGPT-6 (Pach, Tardos, Zeng)","deck":null,"takeaway":"János Pach, one of the conjecture's authors, with Gábor Tardos and Ji Zeng, writes that ChatGPT-6 proved that graphs drawn with no k pairwise crossing edges have linearly many edges, a conjecture known only for k ≤ 4. Their note gives a self-contained combinatorial version; an independent human preprint by Ankan Sadhu (1 Nov) claims a similar proof.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Pach, Tardos, Zeng: On the number of edges in quasiplanar graphs (arXiv 2611.07314)","url":"https://arxiv.org/abs/2611.07314","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Sadhu: Every k-Quasi-Planar Graph Has Linearly Many Edges (arXiv 2611.01509)","url":"https://arxiv.org/abs/2611.01509","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Fox, Pach, Suk: Quasiplanar graphs, string graphs, and the Erdős–Gallai problem (arXiv 2112.02378)","url":"https://arxiv.org/abs/2112.02378","type":"paper","group":"primary","domain":"arxiv.org"}],"official":3,"filed":"2026-11-08","updated":"2026-11-08","orgs":["OpenAI","Rényi Institute","ISTA"],"title":"Pach–Shahrokhi–Szegedy quasiplanar conjecture: 'ChatGPT produced a proof', say Pach, Tardos and Zeng, who post a simplified version; k-quasiplanar graphs have O_k(n) edges","summary":"arXiv 2611.07314 (János Pach, Gábor Tardos, Ji Zeng; 5 Nov 2026) states that ChatGPT-6 produced a proof of the conjecture of Pach, Shahrokhi and Szegedy that k-quasiplanar topological graphs on n vertices have O_k(n) edges, and presents a simplified, purely combinatorial version. Before this the bound was known only for k ≤ 4. A separate preprint by Ankan Sadhu (arXiv 2611.01509, 1 Nov), which says AI only helped with literature and LaTeX, claims a proof with a similar strategy.","key_facts":["Statement: a k-quasiplanar graph (drawn in the plane with no k pairwise crossing edges) on n vertices has O_k(n) edges; the authors' constants give at most 2^{(2+o(1)) k log₂ k} n edges","Before: known only for k ≤ 4 (Agarwal–Aronov–Pach–Pollack–Sharir, Ackerman–Tardos, Ackerman); for fixed k ≥ 5 the best bound was Fox–Pach–Suk's O_k(n (log n)^{2⌈log₂ k⌉−4})","AI disclosure (abstract): 'ChatGPT produced a proof of this conjecture. Here, we present a simplified, self-contained version of that proof.' Body: 'The argument produced by ChatGPT relies on a paper by Beeker and Lazarovich … we present a self-contained and purely combinatorial version'; acknowledgements: 'an effort to interpret and simplify a proof generated by ChatGPT-6'","Independent claim: Ankan Sadhu, 'Every k-Quasi-Planar Graph Has Linearly Many Edges' (arXiv 2611.01509, 1 Nov); its declaration says GPT-5.6 Sol only helped with literature search and converting the author's handwritten proof to LaTeX. Pach–Tardos–Zeng note it uses 'a strategy very similar to the one initially provided to us by ChatGPT'","Verification status: preprints; the human-simplified version is by leading experts in the area (Pach is a co-author of the conjecture); not refereed"],"key_numbers":[],"tags":["math","combinatorics","geometric-graph-theory","chatgpt","gpt-6"],"science":{"field":"mathematics","subfield":"combinatorial geometry / topological graphs","problem":"Pach–Shahrokhi–Szegedy conjecture: k-quasiplanar graphs have O_k(n) edges","result":"Proof produced by ChatGPT-6; simplified self-contained proof by Pach, Tardos and Zeng","open_since":"1996","ai_system":["ChatGPT-6"],"human_role":"AI produced the proof (relying on Beeker–Lazarovich); human experts rewrote and simplified it","verification":{"key":"expert","label":"Expert-checked","text":"Expert-checked preprint (by the conjecture's co-author); not refereed"},"status":"pending","shock":"A conjecture co-posed by Pach is reported proved by ChatGPT, and Pach himself writes up the simplified proof"},"body_md":"## What happened\n\nOn 5 Nov 2026 János Pach, Gábor Tardos and Ji Zeng posted a short paper. It opens by saying that ChatGPT produced a proof of\nthe conjecture of Pach, Shahrokhi and Szegedy that graphs drawn with no k pairwise crossing edges have linearly many edges. The\nmodel's argument used a topological paper by Beeker and Lazarovich. The three authors rewrote it as a self-contained\ncombinatorial proof. Four days earlier, Ankan Sadhu had posted a proof he describes as his own, with a similar strategy.\n\n## Why it matters\n\nThis is a well-known problem in geometric graph theory, and the experts most associated with it confirm the AI-originated\nargument in their own write-up. Two near-simultaneous proofs of the same conjecture also show how hard it is becoming to\nassign priority.","disputed":[{"label":"Verification","text":"Expert-checked preprint (by the conjecture's co-author); not refereed","sources":[]}],"related":[],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-08","type":"filed","text":"Created (science agent; both PDFs read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-08","run":null,"sources_read":"Science agent; both PDFs read","updated":"2026-11-08","human_review":null,"version":{"date":"2026-11-08"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":189,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":128,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":219,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":158,"in_training_data":false}],"short_url":"https://postcutoff.com/s/quasiplanar-graph-conjecture"}