OpenAI · 157-03 · 2026-09-23 · Combinatorics
A linear list-coloring bound in terms of the Hadwiger number
preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026
Family 157. Counterexamples to the Hadwiger and Colin de Verdière conjectures.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy , where is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound . In the positive direction, every finite nonempty graph satisfies for a universal constant .
We prove that every finite nonempty graph G satisfies for an absolute integer C, where is the largest order of a clique minor. This resolves the Linear List Hadwiger conjecture affirmatively.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 157: Graph coloring, clique minors, and Colin de Verdière invariants. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.