For most graphs H, most H-free graphs have a linear homogeneous set

Erdős and Hajnal conjectured that for every graph H there is a constant ε = ε(H) > 0 such that every graph G that does not have H as an induced subgraph contains a clique or a stable set of order Ω(|V (G)|ε). The conjecture would be false if we set ε = 1; however, in an asymptotic setting, we obtain this strengthened form of Erdős and Hajnal’s conjecture for almost every graph H, and in particular for a large class of graphs H defined by variants of the colouring number. ∗This work was completed while this author was at Durham University, supported by EPSRC, grant EP/G066604/1. He is currently supported by a NWO Veni grant. †This author is partially supported by NSERC and FQRNT grants.