For almost all graphs H, almost all H-free graphs have a linear
homogeneous set
Ross Kang, CWI

There have been various structural attacks upon a conjecture of Erdos
and Hajnal from extremal graph theory. This notorious open problem is
concerned with understanding how the exclusion from G a fixed graph H
as an induced subgraph affects the maximum size of a homogenous set (a
clique or stable set) in G. I will focus on recent asymptotic,
probabilistic formulations of the conjecture, proved using Szemerédi
regularity, one of which is joint work with Colin McDiarmid, Bruce
Reed and Alex Scott.