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