Minimum Weight Flat Antichains of Subsets

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