A continuous analogue of Erdos' k-Sperner theorem

A chain in the unit n-cube is a set C subset of [0,1](n) such that for every x = (x(1),...,x(n)) and y = (y(1),...,y(n)) in C we either have x(i) <= y(i) for all i is an element of [n], or x(i) >= y(i) for all i is an element of [n]. We 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. Given this result, we consider the problem of maximising the n-dimensional Lebesgue measure of a measurable set A subset of [0,1](n) subject to the constraint that it satisfies H-1(A boolean AND C) <= kappa for all chains C subset of [0, 1](n), where kappa is a fixed real number from the interval (0, n]. We show that the measure of A is not larger than the measure of the following optimal set: A(kappa)* ={{x(1),...,x(n)) is an element of [0, 1](n) : n-kappa/2 <= Sigma(n)(i=1) x(i) <= n+kappa/2}. Our result may be seen as a continuous counterpart to a theorem of Erdos, regarding k-Sperner families of finite sets. (C) 2019 Elsevier Inc. All rights reserved.