Tournaments with near-linear transitive subsets

Let H be a tournament, and let ≥ 0 be a real number. We call an “Erdős-Hajnal coefficient” for H if there exists c > 0 such that in every tournament G with |V (G)| > 1 not containing H as a subtournament, there is a transitive subset of cardinality at least c|V (G)| . The Erdős-Hajnal conjecture asserts, in one form, that every tournament H has a positive Erdős-Hajnal coefficient. This remains open, but recently the tournaments with Erdős-Hajnal coefficient 1 were completely characterized. In this paper we provide an analogous theorem for tournaments that have an ErdősHajnal coefficient larger than 5/6; we give a construction for them all, and we prove that for any such tournament H there are numbers c, d such that, if a tournament G with |V (G)| > 1 does not contain H as a subtournament, then V (G) can be partitioned into at most c(log(|V (G)|))d transitive subsets.