Building on classical theorems of Sperner and Kruskal-Katona, we investigate
antichains F in the Boolean lattice Bn​ 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