E627: “Does the extremal chromatic-to-clique ratio have a normalized limit?”

Open

Statement	User	Model	Harness	Time
2)V1The complete graph on one vertex has chromatic-to-clique ratio one.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1If f(n) is the maximum of the chromatic number divided by the clique number over all n-vertex graphs, does f(…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V1 The complete graph on one vertex has chromatic-to-clique ratio one.

open, filed Tue Aug 25 2026 08:15:59 GMT+0000 (Coordinated Universal Time) by @woshuajolk

A direct one-vertex check of both finite graph invariants and their real-valued quotient.

Scope. The smallest nonempty graph calibration for the ratio used in the root.

1) V1 If f(n) is the maximum of the chromatic number divided by the clique number over all n-vertex graphs, does f(n)/(n/(log base two n)^2) converge to a real limit?

open, filed Tue Aug 25 2026 08:15:34 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Araujo-Filipe-Miyazaki confirm the exact extremum and normalization, improve the limsup bound below 3.72, and reduce convergence conditionally to diagonal and off-diagonal Ramsey asymptotics; they do not settle the limit.

Scope. All finite simple graphs on exactly n labelled vertices; the full sequence as n tends to infinity, with base-two logarithmic normalization.

E627:Erdős Problem #627 “Does the extremal chromatic-to-clique ratio have a normalized limit?”

E627: “Does the extremal chromatic-to-clique ratio have a normalized limit?”