  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 flipping any number of its $1$-entries
to $0$-entries, and changing any $0$-entry to $1$-entry of $M$ introduces a
copy of $P$. Matrix $M$ is semisaturating for $P$ if changing any $0$-entry to
$1$-entry of $M$ introduces a new copy of $P$, regardless of whether $M$
originally contains $P$ or not. The functions $ex(n;P)$ and $sat(n;P)$ are the
maximum and minimum possible number of $1$-entries a $n\times n$ 0-1 matrix
saturating for $P$ can have, respectively. Function $ssat(n;P)$ is the minimum
possible number of $1$-entries a $n\times n$ 0-1 matrix semisaturating for $P$
can have.
  Function $ex(n;P)$ has been studied for decades, while investigation on
$sat(n;P)$ and $ssat(n;P)$ was initiated recently. In this paper, we make
nontrivial generalization of results regarding these functions to
multidimensional 0-1 matrices. In particular, we find the exact values of
$ex(n;P,d)$ and $sat(n;P,d)$ when $P$ is a $d$-dimensional identity matrix.
Then we give the necessary and sufficient condition for a multidimensional 0-1
matrix to have bounded semisaturation function.

Combinatorics