{"schema":"postcutoff/event@1","as_of":"2026-10-20T07:28:00+02:00","url":"https://postcutoff.com/e/2026-10-20-borodin-kostochka-threshold-4583-gpt-6-astra/","md":"https://postcutoff.com/e/2026-10-20-borodin-kostochka-threshold-4583-gpt-6-astra/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":"2026-10-20-borodin-kostochka-threshold-4583-gpt-6-astra","date":"2026-10-20","date_precision":"day","short_title":"Borodin–Kostochka conjecture claimed for maximum degree ≥ 4583, with GPT-6 Astra assisting","deck":null,"takeaway":"A preprint posted on Zenodo on Oct 20, 2026 by Lizhong Chen (HKUST) claims the Borodin–Kostochka conjecture (1977) for every graph of maximum degree Δ ≥ 4583: χ ≤ max{ω, Δ − 1}.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Zenodo: The Borodin–Kostochka conjecture for maximum degree at least 4583 (L. Chen)","url":"https://zenodo.org/records/23487163","type":"paper","group":"primary","domain":"zenodo.org"},{"n":2,"title":"arXiv 2603.16670: Liu, Sun, Wang, Zeng, On the Borodin–Kostochka conjecture for graphs with large maximum degree","url":"https://arxiv.org/abs/2603.16670v3","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["Hong Kong University of Science and Technology","OpenAI"],"title":"Borodin–Kostochka conjecture claimed for maximum degree ≥ 4583 (down from 5.3 million), with GPT-6 Astra assisting","summary":"A preprint posted on Zenodo on Oct 20, 2026 by Lizhong Chen (HKUST) claims the Borodin–Kostochka conjecture (1977) for every graph of maximum degree Δ ≥ 4583: χ ≤ max{ω, Δ − 1}. The best published threshold was 5.3 × 10⁶ (Liu, Sun, Wang and Zeng, 2026). Its \"Declaration of AI use\" says OpenAI GPT-6 Astra assisted with the resampling estimates, moment-bound optimisation, local-lemma parameter choices and small-deficit constructions. Unrefereed; the conjecture stays open for 9 ≤ Δ < 4583.","key_facts":["Conjecture (Borodin & Kostochka, JCTB 1977): every graph with Δ ≥ 9 has χ ≤ max{ω, Δ − 1}, one colour fewer than Brooks's theorem when there is no clique of size Δ","Earlier: Reed proved it for sufficiently large Δ (JCTB 1999); Liu, Sun, Wang and Zeng gave the explicit threshold 5.3 × 10⁶ (arXiv 2603.16670, v3 May 2026); Chen's own SSRN working manuscript had 2.9 × 10⁶","New claim: Δ ≥ 4583, about a thousandfold lower; method: clique/sparse decomposition, random partial colouring with pre-selected families of witness triples, two applications of the local lemma, finite exponential-moment bounds","The numerical part checks finitely many scalar inequalities in exact rational arithmetic (no graph enumeration); programs, parameters and reference output accompany the paper","Disclosure (verbatim): 'OpenAI GPT-6 Astra assisted with the resampling estimates in Section 6.4, moment-bound optimisation in Sections 7.3 and 8.1, weighted selection and local-lemma parameter choices in Sections 5 and 9, and the small-deficit constructions in Appendix A.3.'","Single author; ~22,800-word PDF; no Lean formalisation; no expert reaction found on Oct 20"],"key_numbers":[],"tags":["mathematics","graph-colouring","ai-assisted-proof","gpt-6-astra","zenodo","preprint"],"science":{"field":"mathematics","subfield":"graph theory (graph colouring)","problem":"Borodin–Kostochka conjecture: χ ≤ max{ω, Δ − 1} for Δ ≥ 9","result":"Claimed proof for all graphs with Δ ≥ 4583, lowering the explicit threshold from 5.3 × 10⁶; the range 9 ≤ Δ < 4583 remains open","open_since":"1977","ai_system":["GPT-6 Astra"],"human_role":"Human-led with AI assistance on specific estimates, parameter optimisation and constructions (author's declaration)","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed Zenodo preprint; rational-arithmetic checks with published code; not peer-reviewed or formalised"},"status":"pending","shock":"","short":"Borodin–Kostochka for Δ ≥ 4583"},"body_md":"## What happened\n\nThe paper does not settle the conjecture, but it claims to move the explicit bound from millions to a few thousand. The AI's role is stated precisely in the paper's own declaration (above). It was posted on\nZenodo, not arXiv, and found through the sweep's Zenodo step; its PDF was read for the disclosure.\n\n## Why it matters\n\nIt is another case of a frontier model credited with the technical estimate-tuning in a probabilistic combinatorics proof on a\nnamed 1977 conjecture. Like other single-author Zenodo claims, it needs expert checking before it counts.","disputed":[{"label":"Verification","text":"Unrefereed Zenodo preprint; rational-arithmetic checks with published code; not peer-reviewed or formalised","sources":[]}],"related":[{"id":"2026-10-19-ai-assisted-results-zenodo-october-17-19","url":"https://postcutoff.com/e/2026-10-19-ai-assisted-results-zenodo-october-17-19/","date":"2026-10-19","date_precision":"day","short_title":"AI-assisted maths on Zenodo, 17–19 Oct 2026","deck":"A claimed proof of the 1985 Arad–Herzog conjecture, Messerschmidt's conjecture confirmed with Codex, and a Claude-run study on AI-written science","takeaway":"Zenodo, not arXiv, carried the weekend's AI-assisted results: an unknown author claims the 40-year-old Arad–Herzog conjecture on products of conjugacy classes with agentic OpenAI and Anthropic tools and a literature-relative Lean check, and a Bologna mathematician confirms Messerschmidt's 2018 conjecture with Codex. None is refereed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":13,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-09-29-list-total-colouring-conjecture-false-astra","url":"https://postcutoff.com/e/2026-09-29-list-total-colouring-conjecture-false-astra/","date":"2026-09-29","date_precision":"day","short_title":"List Total Colouring Conjecture (late 1990s) disproved","deck":"Counterexample found by ChatGPT 6 Astra Ultra 'with little input'","takeaway":"This is a named conjecture from the late 1990s, listed on Open Problem Garden, falling to a prompted AI search, with a counterexample small enough to verify by hand.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-05","updated":"2026-10-07","orgs":["OpenAI","University of Victoria"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-20","type":"filed","text":"Created (Zenodo; PDF read with pdftotext)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-20","run":null,"sources_read":"Zenodo; PDF read with pdftotext","updated":"2026-10-20","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":173,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":112,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":203,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":142,"in_training_data":false}],"short_url":null}