A continuous analogue of Erdős' k-Sperner theorem

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

“…A particular line of research is driven by the idea that several results from extremal combinatorics have continuous counterparts. This is an idea that goes back to the 1970's (see [17]) and, since its conception, has resulted in reporting several analogues of results from extremal combinatorics both in a "measure-theoretic context" (see, for example, [14] and [16]) as well as in a "vector space context" (see, for example, [2], [11] and [15]). In this note, we report yet another measure-theoretic analogue of a result from extremal combinatorics.…”

“…In this article, we investigate a continuous analogue of Theorem 1.1. There are several ways to consider Theorem 1.1 in a continuous setting (see [16] for an alternative direction), but the main idea is to examine what happens when one replaces the binary n-cube {0, 1} n with the unit n-cube [0, 1] n in Theorem 1.1. What is the maximum "size" of a k-antichain in the unit n-cube [0, 1] n ?…”

“…As mentioned in the introduction, there are several ways to consider Theorem 1.1 in a continuous setting, and an alternative direction has been considered in [16]. It is shown in [16] that given s ∈ [0, 1] and β ≥ 0, there exists a set A ⊂ [0, 1] n that satisfies dim H (A) = n − 1 + s and H s (A ∩ C) ≤ β, for all chains C ⊂ [0, 1] n . Here dim H (•) denotes the Hausdorff dimension (see [9, p. 86]).…”

“…The case s = 0, β ∈ N has been the content of the present article. The case s = 1, β ∈ (0, n] has been considered in [16]. The problem remains open for all other values of the parameters s, β.…”