Projections of antichains

TL;DR

This paper characterizes the minimal gap between the size of a weak antichain in integer lattice space and the sum of its projections, providing explicit minimal examples for each size.

Contribution

It explicitly constructs weak antichains that minimize the projection size gap for any given set size in integer lattice space.

Findings

Identifies the minimal gap for weak antichains of any size.

Provides explicit constructions of minimal-gap weak antichains.

Shows sets of a specific form attain the minimum gap.

Abstract

A subset $A$ of $\mathbb{Z}^n$ is called a weak antichain if it does not contain two elements $x$ and $y$ satisfying $x_i<y_i$ for all $i$. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain $A$, the sum of the sizes of its $(n-1)$-dimensional projections must be at least as large as its size $|A|$. They asked what the smallest possible value of the gap between these two quantities is in terms of $|A|$. We answer this question by giving an explicit weak antichain attaining this minimum for each possible value of $|A|$. In particular, we show that sets of the form $A_N=\{x\in\mathbb{Z}^n: 0\leq x_j\leq N-1$ for all $j$ and $x_i=0$ for some $i\}$ minimise the gap among weak antichains of size $|A_N|$.