A continuous analogue of Erdős' \(k\)-Sperner theorem (Q2287347)

The authors show that the \(1\)-dimensional Hausdorff measure of a chain in the unit \(n\)-cube is at most \(n\), and that the bound is sharp. Then they obtain an upper bound for a Lebesgue measure maximization problem with constrains involving such chains, as a continuous counterpart to a theorem of Erdős regarding \(k\)-Sperner families of finite sets.