关于闭Ramsey数$R^{cl}(ω+n,3)$
On the closed Ramsey numbers $R^{cl}(ω+n,3)$
摘要：
本文对可数序数对的拓扑划分关系进行了研究，证明了：对所有整数$n \geq 3$，\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*}，其中$R^{cl}(\cdot,\cdot)$和$R(\cdot,\cdot)$分别表示闭Ramsey数和经典Ramsey数.我们还建立了以下渐近较弱的上界\[ R^{cl}(ω+n,3) \leq ω^2 \cdot n + ω\cdot (n^2-4)+1\]消除使用拉姆齐数。这些结果改进了以前已知的上界和下界。
英文摘要：
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}(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.
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