Upper Bounds for Erdös-Hajnal Coefficients of Tournaments

A celebrated unresolved conjecture of Erdös and Hajnal (see Discrete Appl Math 25 (1989), 37–52) states that for every undirected graph H, there exists ε(H)>0, such that every graph on n vertices which does not contain H as an induced subgraph contains either a clique or an independent set of size at least nε(H). In (Combinatorica (2001), 155–170), Alon et al. proved that this conjecture was equivalent to a similar conjecture about tournaments. In the directed version of the conjecture cliques and stable sets are replaced by transitive subtournaments. For a fixed undirected graph H, define ξ(H) to be the supremum of all ε for which the following holds: for some n0 and every n>n0 every undirected graph with n≥n0 vertices not containing H as an induced subgraph has a clique or independent set of size at least nε. The analogous definition holds if H is a tournament. We call ξ(H) the Erdös–Hajnal coefficient of H. The Erdös–Hajnal conjecture is true if and only if ξ(H)>0 for every H. We prove in this article that: the Erdös–Hajnal coefficient of every graph H is at most 4|H|, there exists η>0 such that the Erdös–Hajnal coefficient of almost every tournament T on k vertices is at most 4k(1+ηlog(k)k), i.e. the proportion of tournaments on k vertices with the coefficient exceeding 4k(1+ηlog(k)k) goes to 0 as k goes to infinity.