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