---
id: "2026-10-06-openai-hadwiger-conjecture-counterexample-claim"
url: "https://postcutoff.com/e/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Awaiting review]
verification: Peer-reviewed
sources: 13
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 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/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/

# Hadwiger's conjecture disproved? (OpenAI, Lean)

Full title: OpenAI's math release claims counterexamples to Hadwiger's conjecture (1943), even in its fractional form, using graphs with independence number at most two; the ordinary disproof is formalized in Lean

"A counterexample to Hadwiger's conjecture" (OpenAI, 23 Sept 2026, 104 pages; family 157 of the 6 Oct release) states: "We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number." For m-vertex examples it proves h(G) < 26m/75 + 2/3 < m/2 ≤ χ_f(G) ≤ χ(G). Companion manuscripts in the family disprove the Colin de Verdière chromatic conjecture and prove the linear list-Hadwiger bound χ_list(G) ≤ C·h(G). A Lean formalization of the ordinary counterexample was added in OpenAI's 7 Oct update and accepted in the oai-math-revised comparator run (25–26 Oct); the fractional and Colin de Verdière statements are outside it. The construction is random and no explicit graph has been published.

## Key facts
- Manuscript (abstract, verbatim): 'We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.' 104 pages, dated 23 Sept 2026
- Mechanism (paper, Theorem 1.1): arbitrarily large m-vertex graphs with α(G) ≤ 2 and connected matching number below m/100; a counting inequality gives h(G) ≤ (m + 4·cm(G) + 2)/3, while α(G) ≤ 2 forces χ_f(G) ≥ m/2. Corollary 1.2: h(G) < 26m/75 + 2/3 < m/2 ≤ χ_f(G) ≤ χ(G). By these bounds the chromatic number exceeds the Hadwiger number by a factor tending to 75/52 ≈ 1.44
- Background: Hadwiger conjectured in 1943 that every graph with no K_t minor is (t−1)-colourable; Bollobás, Catlin and Erdős called it 'one of the deepest unsolved problems in graph theory'. It was known for t ≤ 6 (Robertson–Seymour–Thomas, 1993) and open for all larger t (Wikipedia). Its odd-minor strengthening was disproved by Kühn, Sauermann, Steiner and Wigderson (arXiv 2512.20392, Dec 2025), and the linear relaxation χ ≤ C·t was proved on 4 Oct 2026 by Norin and Steiner with a proof found by GPT-6 Astra (see [2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra](https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/index.md))
- Companions in family 157 (all dated 23 Sept): 'A counterexample to the Colin de Verdière chromatic conjecture' (131 pages; χ exceeds the Colin de Verdière invariant by more than one, again with α ≤ 2) and 'A linear list-coloring bound in terms of the Hadwiger number' (42 pages; 'This resolves the Linear List Hadwiger conjecture affirmatively'). Gu and Xu (arXiv 2610.01946v2) say a key new ingredient of OpenAI's list-colouring proof is a linear choice-number bound for small K_t-minor-free graphs, and give a shorter proof of a weaker form of it
- Lean: OpenAI's initial 6 Oct release formalized only the list-colouring bound (ListHadwiger). The challenge HadwigerCounterexample was added in the 7 Oct update (commit 3014888, pushed 8 Oct 05:03 UTC); the scope note says it proves h(G) < 26m/75 + 2/3 < m/2 ≤ χ(G) for arbitrarily large graphs with α ≤ 2, which 'disproves the ordinary chromatic form', with the fractional and Colin de Verdière statements 'outside this selected statement'. Both challenges were 'accepted' in the oai-math-revised comparator run of 25–26 Oct
- Independent formal work: a Lean repository by the GitHub account EdouardBonnet vendors the release's construction and also derives the fractional consequence h(G) < 26m/75 + 2/3 < m/2 ≤ χ_f(G) ≤ χ(G); its README reports a complete local kernel replay on 11 Oct (Lax archive submission lax-342547)
- No explicit witness: Carlos Toledo's audit lists the Hadwiger graphs among 15 headline objects that are 'random or sufficiently large constructions given only by a prescription', adding: 'Where the Lean is kernel-checked (Hadwiger, Sidorenko, Ryser) the existence is formally proved; the gap is that nobody can look at the object.' An AI-assisted survey (Ying Xie, arXiv 2610.17090) rates the method as new and notes the argument gives no explicit graph
- Reactions: Gil Kalai (7 Oct): 'The new announcement gives counterexamples to the original conjecture, even among graphs with independence number at most two.' Wikipedia now says 'In October 2026, a bulk release by OpenAI included a claimed disproof of the conjecture.' No assessment by a structural graph theorist found (web, Hacker News, oai-math-revised notes) as of 29 Oct

## What happened

Hadwiger's conjecture says that a graph that cannot be contracted to a complete graph on t vertices can be coloured with t − 1
colours. For t = 5 it implies the four colour theorem. It was proved up to t = 6 in 1993 and stayed open beyond. Graphs in which no
three vertices are pairwise non-adjacent (independence number at most two) were a well-studied test case, because there colouring needs
at least half as many colours as there are vertices.

OpenAI's 6 Oct release claims counterexamples in exactly that class. The graphs are built at random so that they have no large
"connected matching", which caps the size of any clique minor at about 35 percent of the vertices, while they still need at least 50
percent as many colours. The Lean proof of this ordinary counterexample was not in the first release. OpenAI added it the next day,
and a community comparator run later accepted it. A volunteer's separate Lean project derives the fractional version as well.

## Why it matters

If the statement holds, one of the best-known conjectures in graph theory is false, and false by a constant factor, not by one colour.
Because the example is random and unpublished, nobody can inspect a concrete graph. The evidence is the kernel-checked existence proof
plus the 104-page argument, which no graph theorist has publicly assessed. The same family also claims the opposite, positive
direction for list colouring, which complements Norin and Steiner's AI-found proof of the linear Hadwiger conjecture two days earlier.

## What is disputed or not yet verified

- Verification: Lean formalization of the ordinary counterexample (added 7 Oct), accepted by an independent comparator run; random construction with no published explicit graph; manuscript unrefereed

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI (PDF): A counterexample to Hadwiger's conjecture (23 Sept 2026)](https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf) (github.com, paper)
2. [OpenAI (PDF): A counterexample to the Colin de Verdière chromatic conjecture](https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf) (github.com, paper)
3. [OpenAI (PDF): A linear list-coloring bound in terms of the Hadwiger number](https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf) (github.com, paper)
4. [OpenAI Lean scope note for family 157](https://github.com/openai/math/blob/main/lean/docs/157.md) (github.com, code)
5. [oai-math-revised comparator results (HadwigerCounterexample, ListHadwiger accepted)](https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv) (github.com, code)
6. [GitHub: EdouardBonnet/hadwiger-counterexample (Lean, fractional consequence)](https://github.com/EdouardBonnet/hadwiger-counterexample) (github.com, code)
7. [arXiv 2610.01946: Gu & Xu, A shorter proof of the linear list Hadwiger conjecture](https://arxiv.org/abs/2610.01946) (arxiv.org, paper)
8. [Wikipedia: Hadwiger conjecture (graph theory)](https://en.wikipedia.org/wiki/Hadwiger_conjecture_(graph_theory)) (en.wikipedia.org, docs)
9. [arXiv 2512.17114: Costa, Luu, Wood & Yip, Verifying Hadwiger's Conjecture for Examples of Graphs with α(G) = 2 (the test case)](https://arxiv.org/abs/2512.17114) (arxiv.org, paper)
10. [arXiv 2512.20392: Kühn, Sauermann, Steiner & Wigderson, Disproof of the Odd Hadwiger Conjecture](https://arxiv.org/abs/2512.20392) (arxiv.org, paper)
11. [Carlos Toledo: OpenAI's math release, what it claims beyond what it checks (finding G10)](https://carlostoledo.co/reports/openai-math.html) (carlostoledo.co, discussion)
12. [Gil Kalai: Updates: Sharing AI progress on mathematics (amazing!) (7 Oct)](https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/) (gilkalai.wordpress.com, discussion)
13. [oai-math-revised AGENT_NOTES for the Hadwiger manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (CONTENTS entry, the three PDFs (104, 131, 42 pp), Lean scope note and its commit history, comparator results, Bonnet's repository, Kalai, Gu–Xu, oai-math-revised notes and Wikipedia read)

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