Pure pairs. IV. Trees in bipartite graphs.

Abstract

In 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.