Towards Erdos-Hajnal for graphs with no 5-hole

TL;DR

This paper advances the understanding of the Erdos-Hajnal conjecture by establishing a stronger bound for graphs that do not contain a 5-cycle, improving previous bounds on clique or independent set sizes.

Contribution

The paper proves a new lower bound for the maximum of clique and independence numbers in C5-free graphs, advancing the Erdos-Hajnal conjecture for this specific case.

Findings

Improved bound: or C5-free graphs, rom 2^{\u221a(\u03bclog n)} to 2^{(\u03bclog n (\u03bclog bclog n))}

Enhanced understanding of structure in graphs without 5-holes

Progress towards the Erdos-Hajnal conjecture for specific graphs

Abstract

The Erdos-Hajnal conjecture says that for every graph $H$ there exists $c>0$ such that $\max(\alpha(G),\omega(G))\ge n^c$ for every $H$-free graph $G$ with $n$ vertices, and this is still open when $H=C_5$. Until now the best bound known on $\max(\alpha(G),\omega(G))$ for $C_5$-free graphs was the general bound of Erdos and Hajnal, that for all $H$, $\max(\alpha(G),\omega(G))\ge 2^{\Omega(\sqrt{\log n })}$ if $G$ is $H$-free. We improve this when $H=C_5$ to $\max(\alpha(G),\omega(G))\ge 2^{\Omega(\sqrt{\log n \log \log n})}.$