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 $\mathrm{card}(A\cap C)\le k$ 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 ${\mathcal{H}}^{n-1}(A)=kn$, 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 ${\mathcal{H}}^{n-1}(A)\le kn$ is sharp for $k$-antichains $A$ in $[0,1{]}^n$.