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Math-2209-A1 · revision r1 · on Math-2209

OpenAI claim summary for Math-2209

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The claimed universal estimate $\chi_{\mathrm{list}}(G)\le Ch(G)$ implies $\chi_{\mathrm{list}}(G)\le C(t-1)$ whenever $K_t$ is excluded as a minor. This covers the stated linear List Hadwiger question with a universal constant, rather than requiring the coefficient-one variant.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. List Hadwiger Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

List Hadwiger Conjecture

OpenAI's claim

Linear list-coloring in clique-minor size: The paper claims a universal constant $C$ with $\chi_{\mathrm{list}}(G)\le C h(G)$ for every finite nonempty graph. The factor is independent of the graph and its order.

Mathematical statement of the claimed result

Mathematical claim 1.
For every finite nonempty simple graph $G$ there is an absolute integer $C\ge1$, independent of $G$, such that \[\chi_{\mathrm{list}}(G)\le C h(G),\qquad h(G)=\max\{t:K_t\text{ is a minor of }G\}.\] The list chromatic number is the least $k$ for which every assignment of $k$ available colors per vertex permits a proper coloring. Consequently $\chi(G)\le Ch(G)$. The constant-factor bound does not imply the conjectured coefficient $1$.

Outline of the argument

Linear list-coloring in clique-minor size: For small minor-free graphs, a random global partition of colors gives each vertex many eligible bins. Stable-set sampling repeatedly reduces fractional chromatic complexity among eligible vertices. If the reduction failed, separator and connectivity arguments would assemble disjoint clique models into the excluded minor; the resulting small-graph bound feeds the global list-coloring argument.

Formal verification

Family 157 describes Lean coverage for the list-coloring upper bound with a universal multiple of clique-minor size; neither chromatic counterexample is part of this selection.

Linear list-coloring in clique-minor size: The manuscript links ListHadwiger.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources