On closed Ramsey numbers of small countable ordinals

Abstract

This paper is a contribution to the investigation of closed partition relations for pairs of countable ordinals. As our main result, we prove that \[ω^4 \cdot (n-2)+1 < R^{cl}(ω\cdot n+1,3)<ω^5\] for every integer $n\ge 3$. This result significantly improves the existing upper and lower bounds for these closed Ramsey numbers. In addition, we prove that \[ω^θ\nrightarrow_{cl} (ω^α,3)^2\] whenever $1\le \alpha \le \theta <{\omega }_1$ satisfy $\theta <R(\alpha ,3)$. This result asymptotically improves the existing lower bounds for $R^{cl}({\omega }^n,3)$ and slightly strengthens the existing necessary condition for being a topological partition ordinal.