Alon–Krivelevich conjecture disproved with GPT-6 Astra
Awaiting review
Importance: major (4 of 5)Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 2 November 2026
Your AI and this story
- GPT-6 Astra183 days after its cutoff
- Claude Opus 5.5122 days after its cutoff
- Gemini 4 Argon213 days after its cutoff
- Grok 4.7152 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 122 days before it.
Key facts
- Construction: for k ≥ 2 and r² > 2(2k−1) ln(2k−1), both vertex classes are copies of all k-subsets of a (2k−1)-set, adjacent when the sets meet in at most r elements; each vertex gets its own subset as list, and the graph is not colourable, so ch(G) ≥ k+1
- Consequence: graphs with ch(G) ≥ (1/32 − o(1))(ln Δ)²/(ln ln Δ)³; the conjecture (Alon and Krivelevich, Annals of Combinatorics 1998) predicted ch = O(log Δ) for bipartite graphs
- Tools: conditional product measures, nested coupling and Doob martingales written as weighted counting, plus Brouwer’s fixed-point theorem
- AI usage declaration (verbatim): ‘We used the gpt-6-astra model through xean [4], a harness for AI-assisted mathematical research, to find the counterexample and its proof. Claude Opus 5.5 was used to rewrite the article. All mathematical content has been independently verified by the authors’
- Verification status: preprint, author-verified; no Lean formalization stated; no outside review seen as of 2 Nov 2026
What happened
Junran Yu (Nanjing Normal University) and Gexin Yu (William & Mary) posted a counterexample to the Alon–Krivelevich conjecture. The bipartite graph has the k-subsets of a (2k−1)-set on both sides, and lists equal to the subsets themselves; the authors show no proper list colouring exists, giving a choice number that grows faster than ln Δ.
Their AI usage declaration says GPT-6 Astra, used “through xean, a harness for AI-assisted mathematical research”, found “the counterexample and its proof”, Claude Opus 5.5 rewrote the article, and the authors verified all content.
Why it matters
The conjecture, from Noga Alon and Michael Krivelevich’s 1998 paper, was a central open question on list colouring of bipartite graphs (the best known upper bound is Δ/log Δ-type). If it holds up, it is one of the more prominent combinatorics conjectures refuted by a model-found construction. We did not see expert reactions yet.
What is disputed or not yet verified
| Verification | Preprint, verified by the authors |
|---|
Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Filed from the 2 Nov sweep (arXiv AI-disclosure list)