  It is well-known that in every $r$-coloring of the edges of the complete
bipartite graph $K_{n,n}$ there is a monochromatic connected component with at
least ${2n\over r}$ vertices. It would be interesting to know whether we can
additionally require that this large component be balanced; that is, is it true
that in every $r$-coloring of $K_{n,n}$ there is a monochromatic component that
meets both sides in at least $n/r$ vertices?
  Over forty years ago, Gy\'arf\'as and Lehel and independently Faudree and
Schelp proved that any $2$-colored $K_{n,n}$ contains a monochromatic $P_n$.
Very recently, Buci\'c, Letzter and Sudakov proved that every $3$-colored
$K_{n,n}$ contains a monochromatic connected matching (a matching whose edges
are in the same connected component) of size $\lceil n/3 \rceil$. So the answer
is strongly "yes" for $1\leq r\leq 3$.
  We provide a short proof of (a non-symmetric version of) the original
question for $1\leq r\leq 3$; that is, every $r$-coloring of $K_{m,n}$ has a
monochromatic component that meets each side in a $1/r$ proportion of its part
size. Then, somewhat surprisingly, we show that the answer to the question is
"no" for all $r\ge 4$. For instance, there are $4$-colorings of $K_{n,n}$ where
the largest balanced monochromatic component has $n/5$ vertices in both partite
classes (instead of $n/4$). Our constructions are based on lower bounds for the
$r$-color bipartite Ramsey number of $P_4$, denoted $f(r)$, which is the
smallest integer $\ell$ such that in every $r$-coloring of the edges of
$K_{\ell,\ell}$ there is a monochromatic path on four vertices. Furthermore,
combined with earlier results, we determine $f(r)$ for every value of $r$.
