A continuous analogue of Erdős' k-Sperner theorem

Abstract A chain in the unit n-cube is a set C ⊂ [ 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 ∈ [ n ] , or x i ≥ y i for all i ∈ [ 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 ⊂ [ 0 , 1 ] n subject to the constraint that it satisfies H 1 ( A ∩ C ) ≤ κ for all chains C ⊂ [ 0 , 1 ] n , where κ 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 κ ⁎ = { ( x 1 , … , x n ) ∈ [ 0 , 1 ] n : n − κ 2 ≤ ∑ i = 1 n x i ≤ n + κ 2 } . Our result may be seen as a continuous counterpart to a theorem of Erdős, regarding k-Sperner families of finite sets.