Ramsey-type theorems with forbidden subgraphs

A graph is called H-free if it contains no induced copy of H. We discuss the following question raised by Erdos and Hajnal. Is it true that for every graph H, there exists an epsilon (H) > 0 such that ally H-free graph with n vertices contains either a complete or an empty subgraph of size at least n(epsilon (H))? We answer this question in the affirmative for a special class of graphs, and give an equivalent reformulation for tournaments. In order to prove the equivalence, we establish several Ramsey type results for tournaments.