  A two-dimensional \emph{grid} is a set $\Gnm = [n]\times[m]$. A grid $\Gnm$
is \emph{$c$-colorable} if there is a function $\chi_{n,m}: \Gnm \to [c]$ such
that there are no rectangles with all four corners the same color. We address
the following question: for which values of $n$ and $m$ is $\Gnm$
$c$-colorable? This problem can be viewed as a bipartite Ramsey problem and is
related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der
Waerden's Theorem). We determine (1) \emph{exactly} which grids are
2-colorable, (2) \emph{exactly} which grids are 3-colorable, and (3)
\emph{exactly} which grids are 4-colorable. We use combinatorics, finite
fields, and tournament graphs.
