Tournaments and the strong Erdős–Hajnal Property

Abstract 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 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 .