Full and maximal squashed flat antichains of minimum weight

A squashed full flat antichain (SFFA) in the Boolean lattice Bn is a family A∪B of subsets of [n] = {1, 2, . . . , n} such that, for some k ∈ [n] and 0 ≤ m ≤ ( n k ) , A is the family of the first m k-sets in squashed (reverse-lexicographic) order and B contains exactly those (k−1)-subsets of [n] that are not contained in some A ∈ A. If, in addition, every k-subset of [n] which is not in A contains some B ∈ B, then A ∪ B is a squashed maximal flat antichain (SMFA). For any n, k and positive real numbers α, β, we determine all SFFA and all SMFA of minimum weight α · |A|+β · |B|. Based on this, asymptotic results on SMFA with minimum size and minimum BLYM value, respectively, are derived.