Pure pairs. VI. Excluding an ordered tree

TL;DR

This paper extends the study of pure pairs in graphs to ordered graphs excluding an ordered forest and its complement, showing that large pure pairs exist with size close to linear, generalizing previous results.

Contribution

It generalizes the existence of large pure pairs in ordered graphs excluding an ordered forest and its complement, providing bounds of size |G|^{1-o(1)}.

Findings

Pure pairs of size |G|^{1-o(1)} exist in ordered graphs excluding an ordered forest and its complement.

Generalizes previous results from monotone paths to all ordered forests.

Shows that linear pure pairs may not always exist, but near-linear size is guaranteed.

Abstract

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 $|G|^{1-o(1)}$.