Largest edge boundary of a down-set
= Largest edge boundary of a down-set
{title2=$h(m)$}
For a <down-set> of size $m$, maximize its <edge boundary> by minimizing the sum of the sizes of its members. Choose all sets in the smallest ranks, followed by any needed part of the next rank; this minimizes that sum among all families and is itself a down-set. With $M_j=\sum_{i=0}^j\binom ni$, $M_{-1}=0$, and $M_{r-1}\le m\le M_r$, the exact maximum is $h(m)=nm-2[\sum_{j<r}j\binom nj+r(m-M_{r-1})]$.