Pivot-minors and the Erdős-Hajnal conjecture

Abstract

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