On the size of maximal antichains and the number of pairwise disjoint maximal chains

Fix integers n and k with n ≥ k ≥ 3. Duffus and Sands proved that if P is a finite poset and n ≤ |C| ≤ n+ (n− k)/(k− 2) for every maximal chain in P , then P must contain k pairwise disjoint maximal antichains. They also constructed a family of examples to show that these inequalities are tight. These examples are 2-dimensional which suggests that the dual statement may also hold. In this paper, we show that this is correct. Specifically, we show that if P is a finite poset and n ≤ |A| ≤ n+ (n− k)/(k− 2) for every maximal antichain in P , then P has k pairwise disjoint maximal chains. Our argument actually proves a somewhat stronger result, and we are able to show that an analogous result holds for antichains.