  Let the product of $t$ chains of equal size be called a $t$-dimensional grid.
The main contribution of this paper is the following theorem: For every
$t$-dimensional grid $P$ there exists a (larger) $t$-dimensional grid $Q$ such
that if we color every comparability of $Q$ with one of $r$ colors, then there
is a subposet $P'$ of $Q$ such that $P\cong P'$, and every comparability of
$P'$ is of the same color.
