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On the closed Ramsey numbers $R^{cl}(ω+n,3)$ (2005.09519v1)

Published 19 May 2020 in math.LO and math.CO

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 \omega2 \cdot n + \omega \cdot (R(n,3)-n)+n\ R{cl}(\omega+n,3) &\leq \omega2 \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 \omega2 \cdot n + \omega \cdot (n2-4)+1] eliminating the use of Ramsey numbers. These results improve the previously known upper and lower bounds.

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