Weighted theorem for k-Sperner families
= Weighted theorem for k-Sperner families
For positive weights $w(i)$ depending only on rank, put $u_i=\binom ni w(i)$. The largest weight of a <k-Sperner family> in $\mathcal P([n])$ is the sum of the $k$ largest $u_i$. Every maximizing family is a union of $k$ full rank levels. To see the equality assertion, its <Lubell mass> must be $k$; decomposing it into $k$ <antichains> by chain height and using <equality in the LYM inequality> forces each part to be a full level.