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

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

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