Pure pairs. VI. Excluding an ordered tree

A pure pair in a graph $G$ is a pair $(Z_1,Z_2)$ of disjoint sets of vertices such that either every vertex in $Z_1$ is adjacent to every vertex in $Z_2$, or there are no edges between $Z_1$ and $Z_2$. With Maria Chudnovsky, we recently proved that, for every forest $F$, every graph $G$ with at least two vertices that does not contain $F$ or its complement as an induced subgraph has a pure pair $(Z_1,Z_2)$ with $|Z_1|,|Z_2|$ linear in $|G|$. Here we investigate what we can say about pure pairs in an {\em ordered} graph $G$, when we exclude an ordered forest $F$ and its complement as induced subgraphs. Fox showed that there need not be a linear pure pair; but Pach and Tomon showed that if $F$ is a monotone path then there is a pure pair of size $c|G|/\log |G|$. We generalise this to all ordered forests, at the cost of a slightly worse bound: we prove that, for every ordered forest $F$, every ordered graph $G$ with at least two vertices that does not contain $F$ or its complement as an induced subgraph has a pure pair of size $|G|^{1-o(1)}$.