A note on the maximum ratio between chromatic number and clique number

TL;DR

This paper improves the upper bound on the maximum ratio between a graph's chromatic number and clique number, refining Erdős's earlier bounds using recent advances in Ramsey number asymptotics.

Contribution

It establishes a new upper bound for the ratio, reducing the constant factor below 3.72, based on recent developments in Ramsey number asymptotics.

Findings

Upper bound for f(n) is now c+o(1) times n/(log n)^2 with c<3.72

First improvement in Erdős's asymptotic bounds for f(n)

Utilizes recent results in Ramsey number asymptotics

Abstract

Let $f(n)$ be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{\chi(G)}{\omega(G)}$, where $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ the clique number of $G$. In 1967, Erd\H{o}s showed that \[ \Big( \frac{1}{4} +o(1) \Big) \frac{n}{(\log_2 n)^2} \le f(n) \le \big( 4+o(1) \big) \frac{n}{(\log_2 n)^2} .\] We show that \[ f(n) \le \big(c+o(1)\big) \frac{n}{(\log_2 n)^2}\] for some $c<3.72$. This follows from recent improvements in the asymptotics of Ramsey numbers and is the first improvement in the asymptotics of $f(n)$ established by Erd\H{o}s.