Summary: 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 \(\beta \rightarrow _{\mathrm{top}}(\alpha ,k)^2\) 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 \(R^{\mathrm{top}}(\alpha ,k)\).
We prove a topological version of the Erdős-Milner theorem, namely that \(R^{\mathrm{top}}(\alpha ,k)\) is countable whenever \(\alpha \) is countable. More precisely, we prove that \(R^{\mathrm{top}}(\omega ^{\omega ^\beta },k+1)\leq \omega ^{\omega ^{\beta \cdot k}}\) for all countable ordinals \(\beta \) and finite \(k\). Our proof is modeled on a new easy proof of a weak version of the 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 \(R^{\mathrm{top}}(\omega +1,k+1)=\omega ^k+1\), \(R^{\mathrm{top}}(\alpha ,k)< \omega ^\omega \) whenever \(\alpha <\omega ^2\), \(R^{\mathrm{top}}(\omega ^2,k)\leq \omega ^\omega \) and \(R^{\mathrm{top}}(\omega ^2+1,k+2)\leq \omega ^{\omega \cdot k}+1\) 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.
For the entire collection see [Zbl 1367.03010].