arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1702.01299math.CO
收藏

关于Erdos和Faudree关于Ramsey数大小的一个问题

On a question of Erdos and Faudree on the size Ramsey numbers

Ramin Javadi, Gholamreza Omidi

备注：
11 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

对于给定的简单图$G_1$和$G_2$，Ramsey数$\hat{R}(G_1,G_2)$是最小的正整数$m$，其中存在一个图$G$，它的$m$边使得在$G$的任意边着色中，红色和蓝色都是$G_1$的红色副本或$G_2$的蓝色副本。1981年，Erdans和Faudree研究了大小Ramsey数$\hat{R}(K_n,tK_2)$，其中$K_n$是$n$顶点上的完全图，而$tK_2$是大小$t$的匹配.当$n\geq 4t-1$和$t=2$时，他们得到了$\hat{R}(K_n,tK_2)$的值，并询问了当$ t $比$ n $大得多时这些数字的行为。在这方面，他们提出了以下有趣的问题：对于每一个正整数$n$，是真的，$$\lim_{t\to \infty} \frac{\hat{R}(K_n,tK_2)} {t\, \hat{R}(K_n,K_2)} = \min\left\{\dfrac{\binom{n+2t-2}{2}} {t\binom{n}{2}}\mid t\in \mathbb{N}\right\} ? $$在本文中，我们得到的确切值$ \hat{R}(K_n,tK_2) $的每一个正整数$ n,t $和作为副产品，我们给一个肯定的答案的问题Erdés和Faudree。

英文摘要：

For given simple graphs $G_1$ and $G_2$, the size Ramsey number $\hat{R}(G_1,G_2)$ is the smallest positive integer $m$, where there exists a graph $G$ with $m$ edges such that in any edge coloring of $G$ with two colors red and blue, there is either a red copy of $G_1$ or a blue copy of $G_2$. In 1981, Erdős and Faudree investigated the size Ramsey number $\hat{R}(K_n,tK_2)$, where $K_n$ is a complete graph on $n$ vertices and $tK_2$ is a matching of size $t$. They obtained the value of $\hat{R}(K_n,tK_2)$ when $n\geq 4t-1$ as well as for $t=2$ and asked for the behavior of these numbers when $ t $ is much larger than $ n $. In this regard, they posed the following interesting question: For every positive integer $n$, is it true that $$\lim_{t\to \infty} \frac{\hat{R}(K_n,tK_2)} {t\, \hat{R}(K_n,K_2)} = \min\left\{\dfrac{\binom{n+2t-2}{2}} {t\binom{n}{2}}\mid t\in \mathbb{N}\right\} ? $$ In this paper, we obtain the exact value of $ \hat{R}(K_n,tK_2) $ for every positive integers $ n,t $ and as a byproduct, we give an affirmative answer to the question of Erdős and Faudree.

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。

↑