An extremal problem for antichains of subsets of a multiset (Q1085160)

Let F be a collection of subsets of a finite set S such that no member of F is contained in any other. Let \(\bar F\) denote the collection of all subsets of S containing a member of F as a subset, and let \b{F} denote the collection of all subsets of S contained in a member of F. Daykin has obtained the best possible upper bound for \(\min(\bar F,\underline F)\), taken over all such F, and has characterized all the extremal F. The author obtains the corresponding upper bound when the set S is replaced by a multiset, and also obtains partial results concerning the extremal cases. The proofs depend on several of the author's results on the generalized Macaulay theorem of Clements and Lindström.