  A $d$-dimensional zero-one matrix $A$ avoids another $d$-dimensional zero-one
matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some
ones to zeroes. Let $f(n,P,d)$ denote the maximum number of ones in a
$d$-dimensional $n \times \cdots \times n$ zero-one matrix that avoids $P$.
  Fox proved for $n$ sufficiently large that $f(n, P, 2) = 2^{k^{\Theta(1)}}n$
for almost all $k \times k$ permutation matrices $P$. We extend this result by
proving for $d \geq 2$ and $n$ sufficiently large that $f(n, P, d) =
2^{k^{\Theta(1)}}n^{d-1}$ for almost all $d$-dimensional permutation matrices
$P$ of dimensions $k \times \cdots \times k$.
