A graph is called H-free if it contains no induced copy of H. We discuss the following question raised by Erdős and Hajnal. Is it true that for every graph H, there exists an \(\) such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least \(\)? 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.

1.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University; Tel-Aviv, Israel; E-mail: noga@math.tau.ac.ilIL

2.Mathematical Institute of the Hungarian Academy of Sciences; H-1364 Budapest, P.O.B. 127, Hungary; E-mail: pach@cims.nyu.eduHU

3.Computer and Automation Institute of the Hungarian Academy of Sciences; H-1518 Budapest, P.O.B. 63, Hungary; E-mail: solymosi@inf.ethz.chHU