Chudnovsky, MariaScott, AlexSeymour, PaulSpirkl, Sophie2023-02-212023-02-212023-01-31https://doi.org/10.1112/plms.12504http://hdl.handle.net/10012/19179© 2023 The Authors. Proceedings of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.The Erdős–Hajnal conjecture says that for every graph H there exists t > 0 such that every graph G not containing H as an induced subgraph has a clique or stable set of cardinality at least IGIt. We prove that this is true when H is a cycle of length five. We also prove several further results: for instance, that if C is a cycle and H is the complement of a forest, there exists t > 0 such that every graph G containing neither of C,H as an induced subgraph has a clique or stable set of cardinality at least IGIt.enAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/Article