# Coordinate deletion of zeroes

## TL;DR

This paper characterizes the optimal structure of subsets of sequences with bounded zeros to minimize their $\,delta$-shadow, providing exact solutions for all sizes.

## Contribution

It proves that sequences with at most r zeros minimize the $\,delta$-shadow and determines the optimal family for every size.

## Key findings

- Sequences with at most r zeros have minimal $\,delta$-shadow.
- The paper provides exact optimal families for all subset sizes.
- The results generalize previous bounds on shadow minimization.

## Abstract

For a family $A\subseteq\left\{ 0,\dots,k\right\} ^{n}$, define the $\delta$-shadow of $A$ to be the set obtained from $A$ by removing from any of its vectors one coordinate that equals zero. Given the size of $A$, how should we choose $A$ to minimise its $\delta$-shadow? Our aim in this paper is to show that, for any $r$, the family of all sequences with at most $r$ zeros has minimal $\delta$-shadow. We actually give the exact best $A$ for every size.