Scott, AlexSeymour, PaulSpirkl, Sophie2023-02-212023-02-212022-09-07https://doi.org/10.1002/jgt.22880http://hdl.handle.net/10012/19177Let H be a tree. It was proved by Rödl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph Kt,t as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Bousquet, Pilipczuk, Rzążewski, Thomassé, and Walczak.enAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/bipartite graphsinduced subgraphsArticle