# On $k$-antichains in the unit $n$-cube

## TL;DR

This paper investigates the size of special subsets called $k$-antichains in the unit $n$-cube, establishing an upper bound on their Hausdorff measure and conjecturing the existence of extremal examples, verified in two dimensions.

## Contribution

It introduces the concept of $k$-antichains in the unit $n$-cube and provides an upper bound on their Hausdorff measure, advancing understanding of their geometric properties.

## Key findings

- Hausdorff measure of $k$-antichains is at most $kn$
- Bound is asymptotically sharp
- Conjecture verified for $n=2$

## Abstract

A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$. We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy \[ \text{card}(A \cap C) \le k, \, \text{ for all chains } \, C \subset [0,1]^n \, , \] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. We show that the $(n-1)$-dimensional Hausdorff measure of a $k$-antichain in $[0,1]^n$ is at most $kn$ and that the bound is asymptotically sharp. Moreover, we conjecture that there exist $k$-antichains in $[0,1]^n$ whose $(n-1)$-dimensional Hausdorff measure equals $kn$ and we verify the validity of this conjecture when $n=2$.