A grid generalisation of the Kruskal-Katona theorem

Abstract

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 $r$.