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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 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. |
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