  A coloring of a complete bipartite graph is shuffle-preserved if it is the
case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the
same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced
subgraph with respect to color $c$ is complete.) In this paper, we investigate
a variant of the Ramsey problem for the class of complete bipartite
multigraphs. (By a multigraph we mean a graph in which multiple edges, but no
loops, are allowed.) Unlike the conventional m-coloring scheme in Ramsey theory
which imposes a constraint (i.e., $m$) on the total number of colors allowed in
a graph, we introduce a relaxed version called m-local coloring which only
requires that, for every vertex $v$, the number of colors associated with $v$'s
incident edges is bounded by $m$. Note that the number of colors found in a
graph under $m$-local coloring may exceed m. We prove that given any $n \times
n$ complete bipartite multigraph $G$, every shuffle-preserved $m$-local
coloring displays a monochromatic copy of $K_{p,p}$ provided that $2(p-1)(m-1)
< n$. Moreover, the above bound is tight when (i) $m=2$, or (ii) $n=2^k$ and
$m=3\cdot 2^{k-2}$ for every integer $k\geq 2$. As for the lower bound of $p$,
we show that the existence of a monochromatic $K_{p,p}$ is not guaranteed if
$p> \lceil \frac{n}{m} \rceil$. Finally, we give a generalization for
$k$-partite graphs and a method applicable to general graphs. Many conclusions
found in $m$-local coloring can be inferred to similar results of $m$-coloring.
