论文标题
在关闭的Ramsey数字上$ r^{cl}（ω+n，3）$
On the closed Ramsey numbers $R^{cl}(ω+n,3)$
论文作者
论文摘要
在本文中，我们有助于研究成对可数序的拓扑分区关系，并证明，对于所有整数$ n \ geq 3 $，\ begin {align*} r^{cl} r^{cl+n，3，3） 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.我们还建立了以下渐近弱的上限\ [r^{cl}（ω+n，3）\leqΩ^2 \ cdot n+ω\ cdot（n^2-4）+1 \]消除了Ramsey数字的使用。这些结果改善了先前已知的上限和下限。
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.
