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

We prove that for every graph $H$, there exists $\epsilon >0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }\ge \epsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erd\H{o}s-Hajnal conjecture for vertex-minors.