Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Strengthening of the Erd\H{o}s-Szekeres Theorem   The Erd\H{o}s-Szekeres Theorem stated in terms of graphs says that any
red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$
contains a red copy of the monotone increasing path with $r$ edges or a blue
copy of the monotone increasing path with $s$ edges. Although $rs + 1$ is the
minimum number of vertices needed for this result, not all edges of $K_{rs+1}$
are necessary. We characterize the subgraphs of $K_{rs+1}$ with this coloring
property as follows: they are exactly the subgraphs that contain all the edges
of a graph we call the circus tent graph $CT(r,s)$.
  Additionally, we use similar proof techniques to improve upon some of the
bounds on the online ordered size Ramsey number of a path given by
P\'erez-Gim\'enez, Pralat, and West.

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The categories of this paper are: plasm-ph
3
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: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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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: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: eess