A minimization problem concerning subsets of a finite set

AbstractLet F be the set of subsets of a finite set S, and for H ⊂ F, let H′ denote the elements of F which are contained in some element of H. Given integers ml and ml+1 does there exist a subset H of F consisting of exactly ml l-element subsets of S and ml+1 (l+1)-element subsets of S such that no two elements of H are related by set-wise inclusion, and if such sets H do exist what the smallest |(l−1)(H′)| can be, where |(l−1)(H′)| is the number of (l−1)-element subsets of S in H′? A generalization of this problem, which was posed by G. Katona, is solved in this paper with the help of the generalized Macaulay theorem [2]