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 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.