Excluding Hooks and their Complements

The long-standing Erdős-Hajnal conjecture states that for every n-vertex undirected graph H there exists ϵ(H)>0 such that every graph G that does not contain H as an induced subgraph contains a clique or an independent set of size at least nϵ(H). A natural weakening of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both H and its complement Hc. These conjectures have been shown to hold for only a handful of graphs: it is not even known if they hold for all graphs on 5vertices.In a recent breakthrough, the symmetrized version of the Erdős-Hajnal conjecture was shown to hold for all paths. The goal of this paper is to show that the symmetrized conjecture holds for all trees on 6(or fewer) vertices. In fact this is a consequence of showing that the symmetrized conjecture holds for any path with a pendant edge at its third vertex; thus we also give a new infinite family of graphs for which the symmetrized conjecture holds.