Towards Erdos-Hajnal for graphs with no 5-hole

Abstract

The Erdos-Hajnal conjecture says that for every graph $H$ there exists $c>0$ such that $\max (\hat{I}\pm (G),\ddot{I}‰(G))\ge n^c$ 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 $\max (\hat{I}\pm (G),\ddot{I}‰(G))$ for $C_5$-free graphs was the general bound of Erdos and Hajnal, that for all $H$, $\max (\hat{I}\pm (G),\ddot{I}‰(G))\ge 2^{\hat{I}\text{c◯}(\sqrt{\log n})}$ if $G$ is $H$-free. We improve this when $H=C_5$ to $\max (\hat{I}\pm (G),\ddot{I}‰(G))\ge 2^{\hat{I}\text{c◯}(\sqrt{\log n\log \log n})}.$