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

A 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 $$ \text{card}(A\cap C)\le k,\quad\text{for all chains }C\subset[0,1]^n, $$ 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 $n=2$.