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 \(P_5\)-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 Erdős-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.