we-solve (beta)

CS-164-A1 · revision r1 · on CS-164

OpenAI claim summary for CS-164

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: 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 result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 106; 2026-09-24).

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

Manuscript

Contents
  1. Independent-set hardness with a three-colorability promise
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Independent-set hardness with a three-colorability promise

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 106; 2026-09-24), the asserted result is that three-colorable graphs are NP-hard to distinguish from graphs of any fixed sufficiently small independence ratio, implying hardness of finding any fixed larger number of colors under that promise.

Mathematical statement of the claimed result

Manuscript 1.
For each fixed real $0<\delta<1/3$, the claimed deterministic polynomial-time reduction maps every $3$CNF formula $\varphi$ to a nonempty finite simple unweighted graph $G_{\varphi}$ such that \[ \varphi\text{ satisfiable}\Rightarrow\chi(G_{\varphi})\le3, \qquad\varphi\text{ unsatisfiable}\Rightarrow\alpha(G_{\varphi})<\delta|V(G_{\varphi})|. \] Here $\alpha$ is maximum independent-set size. The three-coloring is not supplied as part of the graph, and polynomial degree and constants may depend on $\delta$.

Outline of the argument

For manuscript 1 (family 106; 2026-09-24), vertices encode compatible tuple answers on layers. An independent set yields bounded subchain lists, and distributional alignment chooses a shared layer law so the coefficient test decodes a strategy contradicting the projection gap.

Formal verification

For manuscript 1 (family 106; 2026-09-24), OpenAI lists Lean statements for this assertion: three-colorable graphs are NP-hard to distinguish from graphs of any fixed sufficiently small independence ratio, implying hardness of finding any fixed larger number of colors under that promise.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are IndependentSets.lean.

Their named declarations include OAI.LargeIndependentSets.mainTheoremReal.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources