**Journal of Graph Theory**

# Upper Bounds for Erdös–Hajnal Coefficients of Tournaments

## Abstract

A celebrated unresolved conjecture of Erdös and Hajnal (see Discrete Appl Math 25 (1989), 37–52) states that for every undirected graph *H*, there exists , such that every graph on *n* vertices which does not contain *H* as an induced subgraph contains either a clique or an independent set of size at least . In (Combinatorica (2001), 155–170), Alon et al. proved that this conjecture was equivalent to a similar conjecture about tournaments. In the directed version of the conjecture cliques and stable sets are replaced by transitive subtournaments. For a fixed undirected graph *H*, define to be the supremum of all ε for which the following holds: for some *n*_{0} and every every undirected graph with vertices not containing *H* as an induced subgraph has a clique or independent set of size at least . The analogous definition holds if *H* is a tournament. We call the *Erdös–Hajnal coefficient of H*. The Erdös–Hajnal conjecture is true if and only if for every *H*. We prove in this article that:

- the Erdös–Hajnal coefficient of every graph
*H*is at most , - there exists such that the Erdös–Hajnal coefficient of almost every tournament
*T*on*k*vertices is at most , i.e. the proportion of tournaments on*k*vertices with the coefficient exceeding goes to 0 as*k*goes to infinity.