  A 0-1 matrix $M$ contains a 0-1 matrix pattern $P$ if we can obtain $P$ from
$M$ by deleting rows and/or columns and turning arbitrary 1-entries into 0s.
The saturation function $\mathrm{sat}(P,n)$ for a 0-1 matrix pattern $P$
indicates the minimum number of 1s in a $n \times n$ 0-1 matrix that does not
contain $P$, but changing any 0-entry into a 1-entry creates an occurrence of
$P$. Fulek and Keszegh recently showed that the saturation function is either
bounded or in $\Theta(n)$. Building on their results, we find a large class of
patterns with bounded saturation function, including both infinitely many
permutation matrices and infinitely many non-permutation matrices.
