Pivot-minors and the Erd\H{o}s-Hajnal conjecture

We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erd\H{o}s-Hajnal property. More precisely, for every graph $H$, there exists $\epsilon > 0$ such that every $n$-vertex graph with no pivot-minor isomorphic to $H$ contains two sets $A, B$ of vertices such that $|A|, |B| \ge \epsilon n$ and $A$ is complete or anticomplete to $B$.