Upper Bounds for Erdös-Hajnal Coefficients of Tournaments

A version of the Erdős-Hajnal conjecture for tournaments states that for every tournament H every tournament T that does not contain H as a subtournament, contains a transitive subtournament of size at least n (H) for some (H) > 0, where n is the order of T . For any fixed tournament H we can denote by n0(H) the supremum over all ≥ 0 satisfying the following statement: every tournament T of order n ≥ n0 that does not contain H as a subtournament, contains a transitive subtournament of size at least n . The Erdős-Hajnal conjecture is true iff for every tournament H the limit limn0→∞ n0(H), denoted as ξ(H), is positive. The main goal of this talk is to give the upper bounds for the parameter ξ(H), called by us the Erdős-Hajnal coefficient of a tournament H, for many classes of tournaments H. We will also define tournaments called the pseudocelebrities, discuss their properties and mention some open problems that relate pseudocelebrities to the Erdős-Hajnal conjecture.