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 5 vertices. 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.