Extremal subgraphs of the $d$-dimensional grid graph

For each natural number $n$ we determine, both asymptotically and exactly, the maximum number of edges an induced subgraph of order $n$ of the $d$-dimension a grid graph ${\ints}^d$ can have. The asymptotic bound is obtained by using a theorem Bollob\'{a}s and Thomason, and the exact bound is obtained by induction. This generalizes some earlier results for the case $d=2$ on one hand, and for $n\leq 2^d$ on the other.