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

For each integer k ≥ 3 , we find all maximal intervals I k of natural numbers with the following property: whenever the number of elements in every maximal chain in a finite partially ordered set P lies in I k , then P contains k pairwise disjoint maximal antichains. All such I k are of the form [ n , n + ⌊ n − k k − 2 ⌋ ] , where n ≥ k is an integer.