Pure Pairs VI. Excluding an Ordered Tree.
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A pure pair in a graph G is a pair (Z1,Z2) of disjoint sets of vertices such that either every vertex in Z1 is adjacent to every vertex in Z2, or there are no edges between Z1 and Z2. 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 (Z1,Z2) with |Z1|,|Z2| linear in |G|. Here we investigate what we can say about pure pairs in an 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 generalize 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).