Pure pairs. IV. Trees in bipartite graphs

Abstract

In this paper we investigate the bipartite analogue of the strong Erdos-Hajnal property. We prove that for every forest $H$ and every $\ddot{I}„>0$ there exists $\hat{I}\mu >0$, such that if $G$ has a bipartition $(A,B)$ and does not contain $H$ as an induced subgraph, and has at most $(1-\ddot{I}„)|A|\cdot |B|$ edges, then there is a stable set in $G$ that contains at least $\hat{I}\mu |V_i|$ vertices of $V_i$, for $i=1,2$. No graphs $H$ except forests have this property.

Accepted manuscript; see DOI for journal version