Holes with hats and Erd\H{o}s-Hajnal

A "hole-with-hat" in a graph $G$ is an induced subgraph of $G$ that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the "house" is the smallest, on five vertices. It is not known whether there exists $\epsilon>0$ such that every graph $G$ containing no house has a clique or stable set of cardinality at least $|G|^\epsilon$; this is one of the three smallest open cases of the Erd\H{o}s-Hajnal conjecture and has been the subject of much study. We prove that there exists $\epsilon>0$ such that every graph $G$ with no hole-with-hat has a clique or stable set of cardinality at least $|G|^\epsilon$