Summary: Ramsey-type theorems with
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 (H) > 0
such that any H-free graph with n vertices contains either a complete or an empty subgraph
of size at least n(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.
Given a graph G with vertex set V (G) and edge set E(G), let (G) and (G) denote the size of the
largest independent set (empty subgraph) and the size of the largest clique (complete subgraph) in
G, respectively. A subset U V (G) is called homogeneous, if it is either an independent set or a
clique. Denote by hom(G) the size of the largest homogeneous set in G, i.e., let
hom(G) = max ((G), (G)) .
If H is not an induced subgraph of G, then we say that G is an H-free graph.