AbstractA graphGisH-freeif it has no induced subgraph isomorphic toH. We prove that a$$P_5$$P5-free graph with clique number$$\omega \ge 3$$ω≥3has chromatic number at most$$\omega ^{\log _2(\omega )}$$ωlog2(ω). The best previous result was an exponential upper bound$$(5/27)3^{\omega }$$(5/27)3ω, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for$$P_5$$P5, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for$$P_5$$P5-free graphs, and our result is an attempt to approach that.