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Minimising the total number of subsets and supersets

Dec 2022

Let $\mathcal{F}$ be a family of subsets of a ground set $\{1,\ldots,n\}$with $|\mathcal{F}|=m$, and let $\mathcal{F}^{\updownarrow}$ denote the familyof all subsets of $\{1,\ldots,n\}$ that are subsets or supersets of sets in$\mathcal{F}$. Here we determine the minimum value that$|\mathcal{F}^{\updownarrow}|$ can attain as a function of $n$ and $m$. Thiscan be thought of as a `two-sided' Kruskal-Katona style result. It also gives asolution to the isoperimetric problem on the graph whose vertices are thesubsets of $\{1,\ldots,n\}$ and in which two vertices are adjacent if one is asubset of the other. This graph is a supergraph of the $n$-dimensionalhypercube and we note some similarities between our results and Harper'stheorem, which solves the isoperimetric problem for hypercubes. In particular,analogously to Harper's theorem, we show there is a total ordering of thesubsets of $\{1,\ldots,n\}$ such that, for each initial segment $\mathcal{F}$of this ordering, $\mathcal{F}^{\updownarrow}$ has the minimum possible size.Our results also answer a question that arises naturally out of work of Gerbneret al. on cross-Sperner families and allow us to strengthen one of their mainresults.

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