# A grid generalisation of the Kruskal-Katona theorem

## TL;DR

This paper extends the Kruskal-Katona theorem to a grid setting, identifying extremal subsets that minimize the $d$-shadow for sets within multi-valued coordinate spaces.

## Contribution

It provides an exact characterization of extremal sets minimizing the $d$-shadow in grid-structured spaces, generalizing classical combinatorial results.

## Key findings

- Sets $oxed{[t]_r^n}$ are extremal for fixed $t$.
- Sets with at least $r$ zeros are extremal in the unrestricted case.
- The results generalize the Kruskal-Katona theorem to multi-valued grids.

## 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 $r$.