  For a set $A\subseteq\left[k\right]^{n}=\left\{ 0,\dots,k-1\right\} ^{n}$, we
define the $d$-shadow of $A$ to be the set of points obtained by flipping to
zero one of the non-zero coordinates of some point in $A$. Let
$\left[k\right]_{r}^{n}$ be the set of those points in $\left[k\right]^{n}$
with exactly $r$ non-zero coordinates. Given the size of $A$, how should we
choose $A\subseteq\left[k\right]_{r}^{n}$ so as to minimise the $d$-shadow?
Note that the case $k=2$ is answered by the Kruskal-Katona theorem.
  Our aim in this paper is to give an exact answer to this question. In
particular, we show that the sets $\left[t\right]_{r}^{n}$ are extremal for
every $t$. We also give an exact answer to the 'unrestricted' question when we
just have $A\subseteq\left[k\right]^{n}$, showing for example that the set of
points with at least $r$ zeroes is extremal for every $r$.
