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 mk-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.