 Solution

A k-Sperner family meets every maximal chain in a Boolean lattice in at most $k$ members. The random-chain calculation in part (a) therefore gives the generalized LYM inequality $L(\mathcal{A})\le k$. Define the proportion of rank $i$ that is present byThen ${\sum }_i{x}_i=L(\mathcal{A})\le k$ and $w(\mathcal{A})={\sum }_{i}u(i)x_i$. This reduces the weighted theorem for k-Sperner families to allocating at most $k$ units of mass among ranks, each of capacity one.

To handle ties explicitly, let $t>0$ be the $k$th largest value among the $u(i)$, let $H=\{i:u(i)>t\}$, and put $h=|H|<k$. ThenThe last expression is exactly the sum of the $k$ largest rank weights, with multiplicity. Writing those values in descending order as $u_{(1)},\dots ,u_{(n+1)}$, the answer is