Tournaments and the strong Erdős–Hajnal Property

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős–Hajnal Conjecture, states that for every tournament S there exists ɛ(S)>0 such that if T is an n-vertex tournament that does not contain S as a subtournament, then T contains a transitive subtournament on at least n^ɛ(S) vertices. Let C₅ 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=C₅. Here we prove a strengthening of this result, showing that in every tournament T with no subtorunament isomorphic to C₅ 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.