Tournaments with near-linear transitive subsets

Let H be a tournament, and let ϵ≥0 be a real number. We call ϵ an “Erdős-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 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 Erdős-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.