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

TL;DR

This paper investigates the topological partition relations for pairs of countable ordinals, providing new bounds on closed Ramsey numbers involving ω+n and 3, and simplifies upper bounds without using classical Ramsey numbers.

Contribution

It establishes new bounds for closed Ramsey numbers R^{cl}(ω+n,3), improving previous bounds and removing the dependence on classical Ramsey numbers for asymptotic upper bounds.

Findings

Derived new lower bounds for R^{cl}(ω+n,3)

Established improved upper bounds for R^{cl}(ω+n,3)

Eliminated the need for classical Ramsey numbers in upper bounds

Abstract

In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers $n \geq 3$, \begin{align*} R^{cl}(\omega+n,3) &\geq \omega^2 \cdot n + \omega \cdot (R(n,3)-n)+n\\ R^{cl}(\omega+n,3) &\leq \omega^2 \cdot n + \omega \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 \[ R^{cl}(\omega+n,3) \leq \omega^2 \cdot n + \omega \cdot (n^2-4)+1\] eliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.