On closed Ramsey numbers of small countable ordinals

This paper is a contribution to the investigation of closed partition relations for pairs of countable ordinals. As our main result, we prove that $${\omega }^4\cdot (n-2)+1<R^{cl}(\omega \cdot n+1,3)<{\omega }^5$$ 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 $${\omega }^{\theta }{\nrightarrow }_{cl}({\omega }^{\alpha },3{)}^2$$ 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.