---
id: "2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean"
url: "https://postcutoff.com/e/2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-10-30"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 2
editor: human editor
human_review: null
version: "2026-11-02"
---

As of: 2026-11-09 21:13 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean/

# Alon–Krivelevich conjecture disproved with GPT-6 Astra

Full title: Alon–Krivelevich conjecture on choosability of bipartite graphs disproved; counterexample and proof found by GPT-6 Astra (Junran Yu, Gexin Yu)

Junran Yu and Gexin Yu (arXiv 2610.42094, 30 Oct 2026) construct bipartite graphs with maximum degree Δ and choice number at least (1/32 − o(1))(ln Δ)²/(ln ln Δ)³, so ch(G)/ln Δ is unbounded, disproving the 1998 Alon–Krivelevich conjecture that bipartite graphs are O(log Δ)-choosable. The authors state that GPT-6 Astra, run through the "xean" research harness, found the counterexample and its proof, and that they verified everything. Unrefereed preprint.

## 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 183 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 122 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 213 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 152 days after its cutoff

## Sources

1. [arXiv 2610.42094: A Counterexample to the Alon–Krivelevich Conjecture (Junran Yu, Gexin Yu)](https://arxiv.org/abs/2610.42094) (arxiv.org, paper)
2. [Alon & Krivelevich (1998), The choice number of random bipartite graphs](https://doi.org/10.1007/BF01608526) (doi.org, paper)

## Changes

- 2026-11-02 (filed): Created from the 2 Nov sweep (arXiv AI-disclosure list)