Antichains in the set of subsets of a multiset

AbstractA set F of distinct subsets x of a finite multiset M (that is, a set with several different kinds of elements) is a c-antichain if for no c + 1 elements x0, x1, …, xc of F does x0 ⊂ x1 ⊂ ··· ⊂ xc hold. The weight of F, wF, is the total number of elements of M in the various elements x of F. For given integers f and c, we find min wF, where the minimum is taken over all f-element c-antichains F. Daykin [9, 10] has solved this problem for ordinary sets and Clements [3] has solved it for multisets, but only for c = 1