论文标题
$ k_ {2,2} $的$ m $ -bipittite ramsey号码
The $m$-bipartite Ramsey number of the $K_{2,2}$ versus $K_{6,6}$
论文作者
论文摘要
给定的两分图$ g_1，\ ldots，g_n $，二分ramsey number $ br（g_1，\ ldots，g_n）$是最后一个整数$ b $，以使任何完整的二手图$ k_ {b，b} $带有颜色的颜色，颜色为colors colors $ $ 1,2 $ $ ldots $ ldots，n $ cope a a $ ldots， i \ leq n $）$ g_i $的所有边缘都有颜色$ i $。作为双方拉姆齐数字的另一种观点，在给定的两分图中，$ g_1，\ ldots，g_n $和一个正整数$ m $，$ m $ - $ m $ - bipartite ramsey number $ br_m $ br_m（g_1，\ ldots，g_n）$，将其定义为$ b的$ b $ b k_颜色$ 1,2，\ ldots，n $包含一个$ g_i $（$ 1 \ leq i \ leq n $）的副本，其中$ g_i $的所有边缘均具有颜色$ i $。 $ br_m（g_1，g_2）$的尺寸，其中$ g_1 = k_ {2,2} $和$ g_2 \ in \ {k_ {3,3}，k_ {4,4,4} \} $的每个$ m $ an $ br_m（k_ {2,2}，k_ {5,5}）$的特殊值$ m $，到目前为止已在几篇文章中确定。在本文中，我们计算了$ br_m（k_ {2,2}，k_ {6,6}）$的大小，对于某些$ m \ geq 2 $。
Given bipartite graphs $G_1, \ldots, G_n$, the bipartite Ramsey number $BR(G_1,\ldots, G_n)$ is the last integer $b$ such that any complete bipartite graph $K_{b,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copy of some $G_i$ ($1\leq i\leq n$) where all edges of $G_i$ have colour $i$. As another view of bipartite Ramsey numbers, for given bipartite graphs $G_1, \ldots, G_n$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(G_1, \ldots, G_n)$, is defined as the least integer $b$, such that any complete bipartite graph $K_{m,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copy of some $G_i$ ($1\leq i\leq n$) where all edges of $G_i$ have colour $i$. The size of $BR_m(G_1, G_2)$, where $G_1=K_{2,2}$ and $G_2\in \{K_{3,3}, K_{4,4}\}$ for each $m$ and the size of $BR_m(K_{3,3}, K_{3,3})$ and $BR_m(K_{2,2}, K_{5,5})$ for special values of $m$, have been determined in several article up to now. In this article, we compute the size of $BR_m(K_{2,2}, K_{6,6})$ for some $m\geq 2$.
