Tournaments with near-linear transitive subsets

Let H be a tournament, and let ε≥0 be a real number. We call ε an "Erdos-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)|^ε. The Erdos-Hajnal conjecture asserts, in one form, that every tournament H has a positive Erdos-Hajnal coefficient. This remains open, but recently the tournaments with Erdos-Hajnal coefficient 1 were completely characterized. In this paper we provide an analogous theorem for tournaments that have an Erdos-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.