  A graph G is H-free if it has no induced subgraph isomorphic to H. We prove
that a $P_5$-free graph with clique number $\omega\ge 3$ has chromatic number
at most $\omega^{\log_2(\omega)}$. The best previous result was an exponential
upper bound $(5/27)3^{\omega}$, due to Esperet, Lemoine, Maffray, and Morel. A
polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds
for $P_5$, which is the smallest open case. Thus there is great interest in
whether there is a polynomial bound for $P_5$-free graphs, and our result is an
attempt to approach that.
