Polynomial bounds for chromatic number. IV. A near-polynomial bound for excluding the five-vertex path

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a P5-free graph with clique number ω ≥ 3 has chromatic number at most ωlog2(ω). The best previous result, due to Esperet, Lemoine, Maffray, and Morel, was the exponential upper bound (5/27)3ω . There is a general conjecture, due to Esperet, that would imply a polynomial upper bound; and this would imply the celebrated Erdős-Hajnal conjecture for P5, which is the smallest open case of the ErdősHajnal conjecture. Thus there is great interest in Esperet’s conjecture for P5-free graphs, and our result is an attempt to approach that.