Tournaments and the Strong Erd\H{o}s-Hajnal Property

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erd\H{o}s-Hajnal Conjecture, states that for every tournament $S$ there exists $\epsilon(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{\epsilon(S)}$ vertices. Let $C_5$ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon-Pach-Solymosi conjecture is known to be true for the case when $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament $T$ with no subtorunament isomorphic to $C_5$ there exist disjoint vertex subsets $A$ and $B$, each containing a linear proportion of the vertices of $T$, and such that every vertex of $A$ is adjacent to every vertex of $B$.