The Erdős-Hajnal Property for Graphs with No Fixed Cycle as a Pivot-Minor

We prove that for every integer $k$, there exists $\varepsilon > 0$ such n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, exist disjoint sets $A,B \subseteq V(G)$ $|A|,|B| \geq \varepsilon n$, and $A$ is either complete or anticomplete $B$. This proves the analog of Erd\H{o}s-Hajnal conjecture class graphs $C_k$.