On the closed Ramsey numbers $R^{cl}(\omega +n,3)$

Abstract

In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers $n\ge 3$, \begin{align*} R^{cl}(ω+n,3) &\geq ω^2 \cdot n + ω\cdot (R(n,3)-n)+n\\ R^{cl}(ω+n,3) &\leq ω^2 \cdot n + ω\cdot (R(2n-3,3)+1)+1 \end{align*} where $R^{cl}(\cdot ,\cdot )$ and $R(\cdot ,\cdot )$ denote the closed Ramsey numbers and the classical Ramsey numbers respectively. We also establish the following asymptotically weaker upper bound \[ R^{cl}(ω+n,3) \leq ω^2 \cdot n + ω\cdot (n^2-4)+1\] eliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.