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

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