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

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