Minimum Weight Flat Antichains of Subsets

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains ℱ in the Boolean lattice B n of all subsets of [n]:={1,2,… ,n} , where ℱ is flat, meaning that it contains sets of at most two consecutive sizes, say ℱ=𝒜∪ℬ , where 𝒜 contains only k -subsets, while ℬ contains only ( k − 1)-subsets. Moreover, we assume 𝒜 consists of the first m k -subsets in squashed (colexicographic) order, while ℬ consists of all ( k − 1)-subsets not contained in the subsets in 𝒜 . Given reals α , β > 0, we say the weight of ℱ is α· |𝒜|+β· |ℬ| . We characterize the minimum weight antichains ℱ for any given n , k , α , β , and we do the same when in addition ℱ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function.