On a conjecture on $k$-antichains in the unit $n$-cube

Abstract

Let $[0, 1]^{n} \subseteq \mathbb{R}^{n}$ be endowed with its pointwise order, and let $k$ be a positive integer. A subset $A$ of $[0, 1]^{n}$ is said to be a \emph{$k$-antichain} if $\operatorname{card}(A \cap C) \leq k$ for each chain $C \subseteq [0, 1]^{n}$. Letting $\mathcal{H}^{m}$ denote the $m$-dimensional Hausdorff outer measure, Pelekis and Vlasák [Publ.\ Math.\ Debrecen, 2020] conjectured that there exists a $k$-antichain $A \subseteq [0, 1]^{n}$ satisfying $\mathcal{H}^{n-1}(A) = k n$, and proved the special case of this conjecture for $n = 2$, whereas Janzer [Mathematika, 2020] proved the $k = 1$ case of Pelekis and Vlasák's conjecture. This conjecture is motivated by a result due to Erdős on $k$-antichains in $\{ 0, 1 \}^{n}$. We prove Pelekis and Vlasák's conjecture in full generality, thus establishing that their upper bound $\mathcal{H}^{n-1}(A) \leq k n$ is sharp for $k$-antichains $A$ in $[0, 1]^{n}$.