English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study the topological version of the partition calculus in the setting ofcountable ordinals. Let $\alpha$ and $\beta$ be ordinals and let $k$ be apositive integer. We write $\beta\to_{top}(\alpha,k)^2$ to mean that, for everyred-blue coloring of the collection of 2-sized subsets of $\beta$, there iseither a red-homogeneous set homeomorphic to $\alpha$ or a blue-homogeneous setof 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. Moreprecisely, we prove that$R^{top}(\omega^{\omega^\beta},k+1)\leq\omega^{\omega^{\beta\cdot k}}$ for allcountable ordinals $\beta$ and finite $k$. Our proof is modeled on a new easyproof of a weak version of the Erd\H{o}s-Milner theorem that may be ofindependent 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 topologicalpigeonhole principle for ordinals, considerations of a tree ordering based onthe Cantor normal form of ordinals, and some ultrafilter arguments.

Topological Ramsey numbers and countable ordinals