论文标题
两种类型的大小拉姆齐号码用于小订单的比赛
Two types of size Ramsey numbers for matchings of small order
论文作者
论文摘要
对于简单的图表$ g $和$ h $，它们的大小ramsey number $ \ hat {r}（g，h）$是$ f $的最小尺寸，因此对于其边缘的任何红蓝色颜色，$ f $都包含红色$ g $或蓝色$ h $。同样，我们可以通过添加必须连接$ f $的先决条件来定义连接的大小ramsey编号$ {\ hat {r}} _ c（g，h）$。在本文中，我们探讨了这些大小Ramsey数字之间的关系，并在某些类别的图表的值中给出了一些结果。我们主要对$ g $是$ 2K_2 $或$ 3K_2 $的情况感兴趣，而$ h $是循环$ C_N $或路径$ np_m $的联合。此外，我们对$ \ hat {r}（tk_2，p_m）$和$ {\ hat {r}} _ c（tk_2，p_m）$的$ \ hat {r}（tk_2，p_m）$的值提高了上限。
For simple graphs $G$ and $H$, their size Ramsey number $\hat{r}(G,H)$ is the smallest possible size of $F$ such that for any red-blue coloring of its edges, $F$ contains either a red $G$ or a blue $H$. Similarly, we can define the connected size Ramsey number ${\hat{r}}_c(G,H)$ by adding the prerequisite that $F$ must be connected. In this paper, we explore the relationships between these size Ramsey numbers and give some results on their values for certain classes of graphs. We are mainly interested in the cases where $G$ is either a $2K_2$ or a $3K_2$, and where $H$ is either a cycle $C_n$ or a union of paths $nP_m$. Additionally, we improve an upper bound regarding the values of $\hat{r}(tK_2,P_m)$ and ${\hat{r}}_c(tK_2,P_m)$ for certain $t$ and $m$.
