Polynomial $\chi$-boundedness for excluding $P_5$

Abstract

Resolving a 1985 open problem of Gy\'arf\'as, we prove that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path $P_5$. Our approach introduces a chromatic density framework involving chromatic quasirandomness and chromatic density increment, which allows us to deduce the desired statement from the Erd\H{o}s-Hajnal result for $P_5$.