A fractal perspective on optimal antichains and intersecting subsets of the unit $n$-cube

An \emph{$n$-cube antichain} is a subset of the unit $n$-cube $[0,1]^n$ that does not contain two elements $\mathbf{x}=(x_1, x_2,\ldots, x_n)$ and $\mathbf{y}=(y_1, y_2,\ldots, y_n)$ satisfying $x_i\le y_i$ for all $i\in \{1,\ldots,n\}$. Using a chain partition of an adequate finite poset we show that the Hausdorff dimension of an $n$-cube antichain is at most $n-1$.We conjecture that the $(n-1)$-dimensional Hausdorff measure of an $n$-cube antichain is at most $n$ times the Hausdorff measure of a facet of the unit $n$-cube and we verify this conjecture for $n=2$ as well as under the assumption that the $n$-cube antichain is a smooth surface. Our proofs employ estimates on the Hausdorff measure of an $n$-cube antichain in terms of the sum of the Hausdorff measures of its injective projections. Moreover, by proceeding along devil's staircase, we construct a $2$-cube antichain whose $1$-dimensional Hausdorff measure equals $2$. Additionally, we discuss a problem with an intersection condition in a similar setting.