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Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such thatfor any red-blue-green coloring of $K_{N,N}$, there is a red matching of sizeat least $k$ in a component, or a blue matching of at least size $l$ in acomponent, or a green matching of size at least $m$ in a component. In thispaper, we determine the exact value of $r(k,l,m)$ completely. Applying atechnique originated by {\L}uczak that applies Szemer\'edi's Regularity Lemmato reduce the problem of showing the existence of a monochromatic cycle to showthe existence of a monochromatic matching in a component, we obtain the3-colored asymmetric bipartite Ramsey number of cycles asymptotically.

3-colored asymmetric bipartite Ramsey number of connected matchings and cycles