Antichains in the set of subsets of a multiset

A 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, …, x_c of F does x₀ ? x₁ ? ··· ? x_c 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.