  We look at several saturation problems in complete balanced blow-ups of
graphs. We let $H[n]$ denote the blow-up of $H$ onto parts of size $n$ and
refer to a copy of $H$ in $H[n]$ as 'partite' if it has one vertex in each part
of $H[n]$. We then ask how few edges a subgraph $G$ of $H[n]$ can have such
that $G$ has no partite copy of $H$ but such that the addition of any new edge
from $H[n]$ creates a partite $H$. When $H$ is a triangle this value was
determined by Ferrara, Jacobson, Pfender, and Wenger. Our main result is to
calculate this value for $H=K_4$ when $n$ is large. We also give exact results
for paths and stars and show that for $2$-connected graphs the answer is linear
in $n$ whilst for graphs which are not $2$-connected the answer is quadratic in
$n$. We also investigate a similar problem where $G$ is permitted to contain
partite copies of $H$ but we require that the addition of any new edge from
$H[n]$ creates an extra partite copy of $H$. This problem turns out to be much
simpler and we attain exact answers for all cliques and trees.
