Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topological Ramsey numbers and countable ordinals   We study the topological version of the partition calculus in the setting of
countable ordinals. Let $\alpha$ and $\beta$ be ordinals and let $k$ be a
positive integer. We write $\beta\to_{top}(\alpha,k)^2$ to mean that, for every
red-blue coloring of the collection of 2-sized subsets of $\beta$, there is
either a red-homogeneous set homeomorphic to $\alpha$ or a blue-homogeneous set
of size $k$. The least such $\beta$ is the topological Ramsey number
$R^{top}(\alpha,k)$.
  We prove a topological version of the Erd\H{o}s-Milner theorem, namely that
$R^{top}(\alpha,k)$ is countable whenever $\alpha$ is countable. More
precisely, we prove that
$R^{top}(\omega^{\omega^\beta},k+1)\leq\omega^{\omega^{\beta\cdot k}}$ for all
countable ordinals $\beta$ and finite $k$. Our proof is modeled on a new easy
proof of a weak version of the Erd\H{o}s-Milner theorem that may be of
independent interest.
  We also provide more careful upper bounds for certain small values of
$\alpha$, proving among other results that $R^{top}(\omega+1,k+1)=\omega^k+1$,
$R^{top}(\alpha,k)< \omega^\omega$ whenever $\alpha<\omega^2$,
$R^{top}(\omega^2,k)\leq\omega^\omega$ and
$R^{top}(\omega^2+1,k+2)\leq\omega^{\omega\cdot k}+1$ for all finite $k$.
  Our computations use a variety of techniques, including a topological
pigeonhole principle for ordinals, considerations of a tree ordering based on
the Cantor normal form of ordinals, and some ultrafilter arguments.

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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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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: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: atom-ph