arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1503.00937math.CO
收藏

一致超图中松圈的对角Ramsey数

Diagonal Ramsey numbers of loose cycles in uniform hypergraphs

Gholamreza Omidi, Maryam Shahsiah

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

一个$k$ -一致松圈$\mathcal{C}_n^k$是一个顶点集为$\{v_1,v_2,\ldots,v_{n(k-1)}\}$，边集为$e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots,v_{(i-1)(k-1)+k}\}$，$1\leq i\leq n$的超图，这里我们使用模$n(k-1)$算法. Ramsey数$R(\mathcal{C}^k_n,\mathcal{C}^k_n)$是渐近的$\frac{1}{2}(2k-1)n$，正如Gyárfás、Sárközy和Szemerédi所证明的。本文研究了$\mathcal{C}^k_n$的对角Ramsey数的确定问题，证明了对于$n\geq 2$和$k\geq 8$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_n)=(k-1)n+\lfloor\frac{n-1}{2}\rfloor.$$

英文摘要：

A $k$-uniform loose cycle $\mathcal{C}_n^k$ is a hypergraph with vertex set $\{v_1,v_2,\ldots,v_{n(k-1)}\}$ and with the set of $n$ edges $e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots,v_{(i-1)(k-1)+k}\}$, $1\leq i\leq n$, where we use mod $n(k-1)$ arithmetic. The Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ is asymptotically $\frac{1}{2}(2k-1)n$ as has been proved by Gyárfás, Sárközy and Szemerédi. In this paper, we investigate to determining the exact value of diagonal Ramsey number of $\mathcal{C}^k_n$ and we show that for $n\geq 2$ and $k\geq 8$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_n)=(k-1)n+\lfloor\frac{n-1}{2}\rfloor.$$

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑