Caterpillars in Erdős-Hajnal
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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 Thomassé [JCTB 2015], who proved the same when T is a path, and a recent theorem of Choromanski, Falik, Liebenau, Patel and Pilipczuk [Electron. J. Combin. 2018], who proved it when T is a "hook".
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