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 $n \geq 3$. 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 $1 \leq \alpha \leq \theta<\omega_1$ 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.