Topological Ramsey numbers and countable ordinals

Abstract

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 Erdős-Milner theorem, namely that $R^{top}(\alpha ,k)$ is countable whenever $\alpha$ is countable. More precisely, we prove that $R^{top}({\omega }^{{\omega }^{\beta }},k+1)\le {\omega }^{{\omega }^{\beta \cdot k}}$ for all countable ordinals $\beta$ and finite $k$. Our proof is modeled on a new easy proof of a weak version of the Erdős-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 $\alpha <{\omega }^2$, $R^{top}({\omega }^2,k)\le {\omega }^{\omega }$ and $R^{top}({\omega }^2+1,k+2)\le {\omega }^{\omega \cdot k}+1$ 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.

Final version

References in corpus (1)

• Infinite closed monochromatic subsets of a metric space

Cited by in corpus (3)

• Universism and Extensions of V
• The Polarised Partition Relation for Order Types
• The closed ordinal Ramsey number $R^{cl}({\omega }^2,3)={\omega }^6$