  Given two graphs $G$ and $H$, let $f(G,H)$ denote the maximum number $c$ for
which there is a way to color the edges of $G$ with $c$ colors such that every
subgraph $H$ of $G$ has at least two edges of the same color. Equivalently, any
edge-coloring of $G$ with at least $rb(G,H)=f(G,H)+1$ colors contains a rainbow
copy of $H$, where a rainbow subgraph of an edge-colored graph is such that no
two edges of it have the same color. The number $rb(G,H)$ is called the {\it
rainbow number of $H$ with respect to $G$}, and simply called the {\it
bipartite rainbow number of $H$} if $G$ is the complete bipartite graph
$K_{m,n}$. Erd\H{o}s, Simonovits and S\'{o}s showed that $rb(K_n,K_3)=n$. In
2004, Schiermeyer determined the rainbow numbers $rb(K_n,K_k)$ for all $n\geq
k\geq 4$, and the rainbow numbers $rb(K_n,kK_2)$ for all $k\geq 2$ and $n\geq
3k+3$. In this paper we will determine the rainbow numbers $rb(K_{m,n},kK_2)$
for all $k\geq 1$.
