On k-antichains in the unit n-cube

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 consider subsets A, of the unit n-cube [0, 1](n), that satisfy card(A boolean AND C) <= k, for all chains C subset of [0, 1](n), where k is a fixed positive integer. We refer to such a set A as a k-antichain. We show that the (n - 1)-dimensional Hausdorff measure of a k-antichain in [0, 1](n) is at most kn and that the bound is asymptotically sharp. Moreover, we conjecture that there exist k-antichains in [0, 1](n) whose (n - 1)-dimensional Hausdorff measure equals kn, and we verify the validity of this conjecture when n = 2.