Caterpillars in Erdős-Hajnal

Abstract Let T be a tree such that all its vertices of degree more than two lie on one path; that is, T is a caterpillar subdivision. We prove that there exists ϵ > 0 such that for every graph G with | V ( G ) | ≥ 2 not containing T as an induced subgraph, either some vertex has at least ϵ | V ( G ) | neighbours, or there are two disjoint sets of vertices A , B , both of cardinality at least ϵ | V ( G ) | , where there is no edge joining A and B. A consequence is: for every caterpillar subdivision T, there exists c > 0 such that for every graph G containing neither of T and its complement as an induced subgraph, G has a clique or stable set with at least | V ( G ) | c vertices. This extends a theorem of Bousquet, Lagoutte and Thomasse [1] , who proved the same when T is a path, and a recent theorem of Choromanski, Falik, Liebenau, Patel and Pilipczuk [2] , who proved it when T is a “hook”.