## Abstract

The Erdős-Hajnal conjecture asserts that for every graph H there is a constant c>0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of cardinality at least |G|^{c}. In this paper, we prove a conjecture of Liebenau and Pilipczuk [10], that for every forest H there exists c>0, such that every graph G with |G|>1 contains either an induced copy of H, or a vertex of degree at least c|G|, or two disjoint sets of at least c|G| vertices with no edges between them. It follows that for every forest H there exists c>0 such that, if G contains neither H nor its complement as an induced subgraph, then there is a clique or stable set of cardinality at least |G|^{c}.

Original language | English (US) |
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Article number | 107396 |

Journal | Advances in Mathematics |

Volume | 375 |

DOIs | |

State | Published - Dec 2 2020 |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

- Erdos-Hajnal conjecture
- Forests
- Induced subgraphs