A class C of graphs has the "strong Erdos-Hajnal property'' if there is a constant c > 0 with the following property. In every graph in C, with n vertices say, there are two disjoint sets of vertices A and B with |A|,|B| > cn, and either every vertex in A is adjacent to every vertex in B, or there are no edges between A and B.

Liebenau and Pilipczuk conjectured that for every tree T, the class of graphs containing neither T nor its complement as induced subgraphs has the strong Erdos-Hajnal property. I will talk about some progress towards this conjecture, and related results.

Joint work with Anita Liebenau, Marcin Pilipczuk, and Paul Seymour.