Pure pairs. IV. Trees in bipartite graphs

TL;DR

This paper explores a bipartite version of the strong Erdős-Hajnal property, establishing conditions under which large stable sets exist in bipartite graphs excluding certain forests, and showing that only forests possess this property.

Contribution

It proves a bipartite analogue of the strong Erdős-Hajnal property for forests and characterizes the graphs with this property as only forests.

Findings

Existence of large stable sets in bipartite graphs excluding forests

Only forests have the bipartite strong Erdős-Hajnal property

Quantitative bounds relating edges and stable set sizes

Abstract

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