A note on matchings and co-matchings in bipartite graphs

A class of bipartite graphs is said to have the strong Erd\H{o}s-Hajnal property if there exists $\varepsilon > 0$ such that every graph $((A, B), E)$ in the class contains a complete or empty induced subgraph with parts $X \subseteq A$, $Y \subseteq B$ where $|X| \ge \varepsilon|A|$ and $|Y| \ge \varepsilon|B|$. Scott, Seymour and Spirkl \cite{scott2023} proved that it is enough to forbid a forest and the bipartite complement of a forest. In this paper, we provide quantitative bounds on $\varepsilon$ when we restrict to matchings.