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CS-164 · computer science · mathematics · Computational complexity · Graph theory and combinatorics

Independent-set hardness with a three-colorability promise

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed $\delta \in (0,1/3)$, can 3SAT be reduced in deterministic polynomial time to a nonempty finite simple unweighted graph so that satisfiable formulas yield three-colorable graphs and unsatisfiable formulas yield graphs with independence number $<\delta $ times the vertex count?

Short English statement

For every fixed $\delta \in (0,1/3)$, can 3SAT be reduced in deterministic polynomial time to a nonempty finite simple unweighted graph so that satisfiable formulas yield three-colorable graphs and unsatisfiable formulas yield graphs with independence number $<\delta $ times the vertex count?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-164-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-164

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).

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

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