Scott, AlexSeymour, PaulSpirkl, Sophie2023-03-312023-03-312023-07https://doi.org/10.1016/j.jctb.2023.02.005http://hdl.handle.net/10012/19242In this paper we investigate the bipartite analogue of the strong Erd˝os-Hajnal property. We prove that for every forest H and every τ with 0 < τ ≤ 1, there exists ε > 0, such that if G has a bipartition (A,B) and does not contain H as an induced subgraph, and has at most (1− τ )|A| · |B| edges, then there is a stable set X of G with |X ∩ A| ≥ ε|A| and |X ∩ B| ≥ ε|B|. No graphs H except forests have this property.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/induced subgraphbipartite graphtreepure pairArticle