{"schema":"postcutoff/event@1","as_of":"2026-11-15T00:33:00+01:00","url":"https://postcutoff.com/e/2026-11-06-k-clique-hypothesis-refuted-vassilevska-williams-zamir/","md":"https://postcutoff.com/e/2026-11-06-k-clique-hypothesis-refuted-vassilevska-williams-zamir/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":"2026-11-06-k-clique-hypothesis-refuted-vassilevska-williams-zamir","date":"2026-11-06","date_precision":"day","short_title":"k-Clique hypothesis refuted (Vassilevska Williams, Zamir, ChatGPT-6)","deck":null,"takeaway":"A central hypothesis of fine-grained complexity said k-Clique cannot be solved faster than about n^(ωk/3). A new preprint gives an n^((ω/3 − c)k) algorithm for every large k, and attributes its key algebraic lemma 'almost-purely' to ChatGPT-6 working from OpenAI's ETH manuscript.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2611.09764: The k-Clique hypothesis is dead, for good (Vassilevska Williams, Zamir)","url":"https://arxiv.org/abs/2611.09764","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2611.05082: Finding hypercliques faster than brute force (Alman, Li, Pant, Pratt, Williams)","url":"https://arxiv.org/abs/2611.05082","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-11-14","updated":"2026-11-14","orgs":["MIT","Tel Aviv University","OpenAI"],"title":"'The k-Clique hypothesis is dead, for good': Vassilevska Williams and Zamir beat n^(ωk/3) for all large k, even if ω = 2, with a decomposition lemma distilled with ChatGPT-6 from OpenAI's ETH refutation","summary":"arXiv 2611.09764 (6 Nov 2026), Virginia Vassilevska Williams and Or Zamir: an algorithm solves (and counts weighted) k-Clique on n-vertex graphs in n^((ω/3 − c)k) time for every large enough k, with c > 0 independent of ω and k. This refutes the k-Clique conjecture regardless of the matrix-multiplication exponent, even if ω = 2, and also speeds up exact Max-2-CSP algorithms. The core 'rank-or-swap' decomposition of the identity matrix is a binary specialization of a construction in OpenAI's ETH-refutation manuscript, distilled with ChatGPT-6; the authors credit it 'almost-purely' to the AI and the algorithmic framework to themselves.","key_facts":["Claim: k-Clique in n^((ω/3 − c)k) time for large k; previous best for decades was the Nešetřil–Poljak n^(ωk/3) via matrix multiplication, and the k-Clique hypothesis (a standard assumption in fine-grained complexity) said this is essentially optimal","AI statement (Section 6, verbatim): 'This work was achieved by a combination of a traditional research process with extensive repeated use of a publicly available AI tool (OpenAI's ChatGPT-6, on the web interface without special harnesses, tools, token use, etc.)'; the model first found an improvement for 5-cliques beyond ω(2,1,2); 'we view the algebraic decomposition of Section 4 as almost-purely the AI's contribution (originating from [Ope26a]), and the algorithmic framework that uses it as a black-box in Section 3 as mainly ours'","Observation by the authors: 'the obfuscated and cumbersome proofs generated by AI are very difficult for us humans to read and comprehend, we were impressed that they are also difficult for the AI itself to understand … when presented with a clean simplification of the ideas, they (as well as us) were suddenly much more able to take further steps'","Lineage: OpenAI's ETH-refutation manuscript of 26 Oct ([Ope26a]); similar ideas appear in three AI-assisted refutations of the k-HyperClique conjecture (Alman, Li, Pant, Pratt, R. Williams, arXiv 2611.05082; Fischer; and a third group), see 2026-11-05-3-uniform-hyperclique-hypothesis-false-chatgpt-sol","Status: unrefereed preprint; no Lean formalization stated; no public expert reaction found by 14 Nov (HN search rate-limited)"],"key_numbers":[],"tags":["computer-science","complexity-theory","fine-grained-complexity","algorithms","k-clique","chatgpt-6","openai-math-release","arxiv"],"science":{"field":"computer-science","subfield":"fine-grained complexity","problem":"k-Clique hypothesis (no n^((ω/3 − ε)k) algorithm for k-Clique)","result":"Refuted: n^((ω/3 − c)k) for all large k with an absolute c > 0, including weighted counting","open_since":"","ai_system":["ChatGPT-6"],"human_role":"AI-assisted: authors supplied the approach and built the algorithmic framework; ChatGPT-6 produced and helped distill the algebraic decomposition","verification":{"key":"preprint","label":"Preprint only","text":"Preprint, unrefereed"},"status":"pending","shock":"A cornerstone hypothesis used for dozens of conditional lower bounds falls, three weeks after an AI-produced ETH refutation supplied the key tool."},"body_md":"## What happened\n\nOn 6 Nov 2026 Virginia Vassilevska Williams (MIT) and Or Zamir (Tel Aviv University) posted an algorithm that finds k-cliques\nfaster than the matrix-multiplication bound n^(ωk/3) by a polynomial factor in the exponent, for every large enough k and for any\nvalue of ω. The k-Clique hypothesis, which says no such algorithm exists, has been one of the standard assumptions of fine-grained\ncomplexity. The key tool is a decomposition of the identity matrix into a low-rank part plus a few \"partial swaps\", which the\nauthors trace to OpenAI's October manuscript refuting the Exponential Time Hypothesis and which they developed with ChatGPT-6.\n\n## Why it matters\n\nAfter the 3SUM/APSP and ETH claims of October and the hyperclique refutations of early November, this removes another basic\nhardness assumption, and the authors give a detailed account of how the model's contribution was found and cleaned up. Like the\nETH claim it builds on, it is not yet refereed.","disputed":[{"label":"Verification","text":"Preprint, unrefereed","sources":[]}],"related":[{"id":"2026-11-14-ai-assisted-results-november-14-listing","url":"https://postcutoff.com/e/2026-11-14-ai-assisted-results-november-14-listing/","date":"2026-11-14","date_precision":"day","short_title":"AI-assisted results, 14 Nov","deck":null,"takeaway":"arXiv posted no new listing after 13 Nov (it announces Sunday to Thursday), so this sweep is mostly Hexagon's 13 Nov deposits, older arXiv papers not yet catalogued, and the week's commentary. Eight items got their own entries, four of them from a backlog of 6–11 Nov arXiv papers never catalogued; the rest are listed here with the disclosure quoted. All are unrefereed.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":13,"filed":"2026-11-14","updated":"2026-11-14","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-11-05-3-uniform-hyperclique-hypothesis-false-chatgpt-sol","url":"https://postcutoff.com/e/2026-11-05-3-uniform-hyperclique-hypothesis-false-chatgpt-sol/","date":"2026-11-05","date_precision":"day","short_title":"3-uniform hyperclique hypothesis refuted with ChatGPT Sol 6.1","deck":null,"takeaway":"The h-uniform hyperclique hypothesis was a standard assumption for conditional lower bounds.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-05","updated":"2026-11-05","orgs":["OpenAI","University of Illinois Urbana-Champaign"]},{"id":"2026-10-26-openai-eth-false-subexponential-sat-claim","url":"https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims ETH is false","deck":null,"takeaway":"Anthropic independently posted a different Lean-checked refutation the next day (separate entry). OpenAI's model output claims a deterministic algorithm that counts the satisfying assignments of any k-CNF in time 2^(n/(log n)^(1/2+o(1))). If correct, it refutes ETH, SETH, Gap-ETH and NSETH, the assumptions behind hundreds of fine-grained hardness results. OpenAI says the main theorem is formalized in Lean. A working group led by ETH's proposers, Russell Impagliazzo and Ramamohan Paturi, with Ryan Williams, will examine the proof. No expert has assessed it in public yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":35,"official":11,"filed":"2026-10-26","updated":"2026-11-08","orgs":["OpenAI"]},{"id":"2026-10-05-claude-refutes-3sum-apsp-hypotheses","url":"https://postcutoff.com/e/2026-10-05-claude-refutes-3sum-apsp-hypotheses/","date":"2026-10-05","date_precision":"day","short_title":"Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses","deck":"Truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams, Lean-verified)","takeaway":null,"category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-06","updated":"2026-10-13","orgs":["Anthropic","Columbia University","MIT"]}],"people":[{"id":"virginia-vassilevska-williams","name":"Virginia Vassilevska Williams","url":"https://postcutoff.com/person/virginia-vassilevska-williams/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-14","type":"filed","text":"Created (science agent)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-14","run":null,"sources_read":null,"updated":"2026-11-14","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":190,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":129,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":220,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":159,"in_training_data":false}],"short_url":"https://postcutoff.com/s/k-clique-hypothesis-refuted"}