论文标题
$ M $ -Bipittite Ramsey编号$ BR_M（H_1，H_2）$
The $m$-bipartite Ramsey number $BR_m(H_1,H_2)$
论文作者
论文摘要
在$（g^1，g^2）$的颜色$ g $中，$ g $的每个边缘都在$ g^1 $或$ g^2 $中。对于两个两部分图$ h_1 $和$ h_2 $，两部分ramsey number $ br（h_1，h_2）$是最小整数$ b \ geq 1 $，因此每$（g^1，g^2）$ colly（g^1，g^2）$ colterage $ k_ k_ b，b，b} $ co^qu^1 qu^1 qu^1 qu^1 $ H_2 \ subseteq g^2 $。作为另一个观点，对于两分图，$ h_1 $和$ h_2 $和一个正整数$ m $，$ m $ - bipartite ramsey number $ br_m（h_1，h_1，h_2）$ $ h_1 $ and $ h_2 $ and $ h_2 $ is $ h_2 $ is $ g $ n $，以至于每一个$ g $ g $ g $ k_1 $ k_______________________________________________________________________ $ h_2 \ subseteq \ overline {g} $。 The size of $m$-bipartite Ramsey number $BR_m(K_{2,2}, K_{2,2})$, the size of $m$-bipartite Ramsey number $BR_m(K_{2,2}, K_{3,3})$ and the size of $m$-bipartite Ramsey number $BR_m(K_{3,3},到目前为止，已经在几篇文章中计算了K_ {3,3}）$。在本文中，我们确定每种$ M \ geq 2 $的$ br_m（k_ {2,2}，k_ {4,4}）$的确切值。
In a $(G^1,G^2)$ coloring of a graph $G$, every edge of $G$ is in $G^1$ or $G^2$. For two bipartite graphs $H_1$ and $H_2$, the bipartite Ramsey number $BR(H_1, H_2)$ is the least integer $b\geq 1$, such that for every $(G^1, G^2)$ coloring of the complete bipartite graph $K_{b,b}$, results in either $H_1\subseteq G^1$ or $H_2\subseteq G^2$. As another view, for bipartite graphs $H_1$ and $H_2$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(H_1, H_2)$ of $H_1$ and $H_2$ is the least integer $n$, such that every subgraph $G$ of $K_{m,n}$ results in $H_1\subseteq G$ or $H_2\subseteq \overline{G}$. The size of $m$-bipartite Ramsey number $BR_m(K_{2,2}, K_{2,2})$, the size of $m$-bipartite Ramsey number $BR_m(K_{2,2}, K_{3,3})$ and the size of $m$-bipartite Ramsey number $BR_m(K_{3,3}, K_{3,3})$ have been computed in several articles up to now. In this paper we determine the exact value of $BR_m(K_{2,2}, K_{4,4})$ for each $m\geq 2$.
