  For posets $P$ and $Q$, extremal and saturation problems about weak and
strong $P$-free subposets of $Q$ have been studied mostly in the case $Q$ is
the Boolean poset $Q_n$, the poset of all subsets of an $n$-element set ordered
by inclusion. In this paper, we study some instances of the problem with $Q$
being the grid, and its connections to the Boolean case and to the forbidden
submatrix problem.
