Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $|G|^{1-o(1)}$.

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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom