Tournaments with near-linear transitive subsets

Let H be a tournament, and let ? ? 0 be a real number. We call ? an "Erd?s-Hajnal coefficient" for H if there exists c 0 such that in every tournament G not containing H as a subtournament, there is a transitive subset of cardinality at least c | V ( G ) | ? . The Erd?s-Hajnal conjecture asserts, in one form, that every tournament H has a positive Erd?s-Hajnal coefficient. This remains open, but recently the tournaments with Erd?s-Hajnal coefficient 1 were completely characterized. In this paper we provide an analogous theorem for tournaments that have an Erd?s-Hajnal 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.