Weighted intersecting family bound on an odd Boolean lattice (${\sum }_{S\in \mathcal{F}}{w}^{|S|}\le {\sum }_{r>n/2}\left(\genfrac{}{}{0ex}{}{n}{r}\right)w^r$)

For odd $n$ and $w\ge 1$, an intersecting family contains at most one set from each complementary pair in the Boolean lattice. Choosing the larger member in every pair gives the displayed bound and is feasible, because all sets of size greater than $n/2$ intersect. For $w>1$ it is the unique maximizer; for $w=1$ other maximizing families can exist.