Minimising the total number of subsets and supersets

Adam Gowty, Daniel Horsley, Adam Mammoliti

Let $\mathcal{F}$ be a family of subsets of a ground set $\{1,\dots ,n\}$ with $|\mathcal{F}|=m$, and let ${\mathcal{F}}^{\updownarrow }$ denote the family of all subsets of $\{1,\ldo…