Georgia Institute of Technology College of Sciences

The celebrated Erdos-Hajnal conjecture states that for every graph H, there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of size at least n^c, where n = |V(G)|. One approach for proving this conjecture is to prove that in every H-free graph G, there are two linear-size sets A and B such that either there are no edges between A and B, or every vertex in A is adjacent to every vertex in B. As is turns out, this is not true unless both H and its complement are trees. In the case when G contains neither H nor its complement as an induced subgraph, the conclusion above was conjectured to be true for all trees (Liebenau & Pilipczuk), and I will discuss a proof of this for a class of tree called "caterpillars". I will also talk about results and open questions for some variants, including allowing one or both of A and B to have size n^c instead of linear size, and requiring the bipartite graph between A and B to have high or low density instead of being complete or empty. In particular, our results improve the bound on the size of the largest clique or stable that must be present in a graph with no induced five-cycle. Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott, and Paul Seymour.