Polynomial bounds for chromatic number VIII. Excluding a path and a complete multipartite graph

Abstract

We prove that for every path $H$, and every integer $d$, there is a polynomial $f$ such that every graph $G$ with chromatic number greater than $f\left(t\right)$ either contains $H$ as an induced subgraph, or contains as a subgraph the complete $d$-partite graph with parts of cardinality $t$. For $t=1$ and general $d$ this is a classical theorem of Gyárfás, and for $d=2$ and general $t$ this is a theorem of Bonamy et al.