Saturation problems about forbidden $0$-$1$ submatrices

A $0$-$1$ matrix $M$ is saturating for a $0$-$1$ matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by changing some $1$ entries to $0$ entries, and changing an arbitrary $0$ to $1$ in $M$ introduces such a submatrix in $M$. In saturation problems for $0$-$1$ matrices we are interested in estimating the minimum number of $1$ entries in an $m \times n$ matrix that is saturating for $P$, in terms of $m$ and $n$. In other words, we wish to give good estimates for the saturation function of $P$. Recently, Brualdi and Cao initiated the study of saturation problems in the context of $0$-$1$ matrices. We extend their work in several directions. We prove that every $0$-$1$ forbidden matrix has its saturation function either in $\Theta(1)$ or $\Theta(n)$ in the case when we restrict ourselves to square saturating matrices. Then we give a partial answer to a question posed by Brualdi and Cao about the saturation function of $J_k$, which is obtained from the identity matrix $I_k$ by putting the first row after the last row. Furthermore, we exhibit a $5\times 5$ permutation matrix with the saturation function bounded from the above by a fixed constant. We complement this result by identifying large classes of $0$-$1$ matrices with linear saturation function. Finally, we completely resolve the related semisaturation problem as far as the constant vs. linear dichotomy is concerned.