Ramsey-type theorems with forbidden subgraphs

A graph is called H-free if it contains no induced copy of H. We discuss th e 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 cl ass of graphs, and give an equivalent reformulation for tournaments. In ord er to prove the equivalence, we establish several Ramsey type results for t ournaments.