Topological partition relations for countable ordinals
Volume 244 / 2019
Fundamenta Mathematicae 244 (2019), 147-166
MSC: Primary 03E02.
DOI: 10.4064/fm413-1-2018
Published online: 24 August 2018
Abstract
Using families of finite sets to represent countable ordinals, we find, for each $k\geq 2$ and each $m$ with $4\leq m\leq 2k+5\binom{k}{2}$, the least ordinal $\gamma$ such that for every $l$ we have $\gamma\xrightarrow[\rm top]{} (\omega\cdot 2+1)^2_{l,m}$ as topological spaces. We also prove that if $\gamma$ is such that for every $l$ the relation $\gamma\xrightarrow[\rm top]{} (\omega\cdot 2+1)^2_{l,3}$ holds, then $\gamma\geq\omega^{\omega^{\omega}}$.
