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 if T an n-vertex does not contain as a subtournament, then contains transitive subtournament on at least nɛ(S) vertices. Let C5 be unique five-vertex where vertex has two inneighbors outneighbors. The Alon–Pach–Solymosi known true case when S=C5. Here we prove strengthening this result, showing in with no subtorunament isomorphic exist disjoint subsets B, each containing linear proportion vertices T, adjacent B.