Scott, AlexSeymour, PaulSpirkl, Sophie2023-12-052023-12-052023-09-15https://doi.org/10.1007/s00493-023-00015-whttp://hdl.handle.net/10012/20133This article is made available open access through a Creative Commons Attribution 4.0 International license http://creativecommons.org/licenses/by/4.0/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 was an exponential upper bound (5/27)3ω, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erd˝os-Hajnal conjecture holds for P5, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P5-free graphs, and our result is an attempt to approach that.enAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/chromatic numberinduced subgraphsArticle