Forcing monochromatic subdivisions

Abstract

We prove that for every integers $d \ge 1$ and $s\ge2$ there exists an integer $D$, that depends only on $d$ and $s$, such that for every graph $P$ with maximum degree at most $ d$, there is a graph $H$ with maximum degree at most $D$ in which every $s$-coloring of $V(H)$ yields a monochromatic subdivision of $P$.