On the maximum ratio between chromatic number and clique number

TL;DR

This work investigates the maximum possible ratio χ(G)/ω(G) over n-vertex graphs, encapsulated in f(n). It forges a direct link between f(n) and Ramsey-number behavior by introducing g(n) and analyzing its liminf/limsup via L = liminf log R(k,k)/k and M = limsup log R(s,t)/√(st), yielding both lower and upper bounds. Unconditionally, the authors prove f(n) ≤ (3.71943+o(1)) n/(log n)^2, and, assuming a weak multicolor Ramsey Diagonal Conjecture, obtain f(n) ≤ (3.70831+o(1)) n/(log n)^2, representing the first improvement on Erdős’s asymptotics. The results hinge on Ramsey-number bounds, a greedy coloring lemma, and entropy-based optimization to bound log R(s,t) and its impact on χ/ω. The findings illuminate a path toward tightening the constant in Erdős’s bound through Ramsey-theoretic advances.

Abstract

Let $f(n)$ be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{χ(G)}{ω(G)}$, where $χ(G)$ denotes the chromatic number of $G$ and $ω(G)$ the clique number of $G$. In 1967, Erdős 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 Erdős.

Theorems & Definitions (12)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• proof
• proof : Proof of \ref{['thm:lower']}
• Lemma 2.4
• proof : Proof of \ref{['thm:upper']}
• proof : Proof of Theorem \ref{['thm:g_upper']}