Upper Bounds for Erdös–Hajnal Coefficients of Tournaments

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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 inline image, 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 inline image. 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 inline image to be the supremum of all ε for which the following holds: for some n0 and every inline image every undirected graph with inline image vertices not containing H as an induced subgraph has a clique or independent set of size at least inline image. The analogous definition holds if H is a tournament. We call inline image the Erdös–Hajnal coefficient of H. The Erdös–Hajnal conjecture is true if and only if inline image for every H. We prove in this article that:

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

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