### 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 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”.

Original language | English (US) |
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Journal | Journal of Combinatorial Theory. Series B |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory. Series B*. https://doi.org/10.1016/j.jctb.2018.09.002