  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
$\max(\alpha(G),\omega(G))\ge 2^{\Omega(\sqrt{\log n \log \log n})}.$
