Liebenau, AnitaPilipczuk, MarcinSeymour, PaulSpirkl, Sophie2022-08-122022-08-122019-05https://doi.org/10.1016/j.jctb.2018.09.002http://hdl.handle.net/10012/18512The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2018.09.002. © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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 [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”.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/induced subgraphscaterpillarsErdős–Hajnal conjectureArticle