Minimising the total number of subsets and supersets

TL;DR

This paper determines the minimal size of the family of all subsets and supersets of a given family of subsets, providing a two-sided Kruskal-Katona type result and insights into an isoperimetric problem on subset graphs.

Contribution

It introduces the minimal size characterization of the union of subsets and supersets of a family, extending Kruskal-Katona and isoperimetric results to a two-sided setting.

Findings

Established the minimum size of the two-sided family as a function of n and m.

Identified a total ordering of subsets minimizing the size of the two-sided family.

Connected the problem to isoperimetric problems on subset graphs and related it to Harper's theorem.

Abstract

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 family of 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 $m$. This can be thought of as a `two-sided' Kruskal-Katona style result. It also gives a solution to the isoperimetric problem on the graph whose vertices are the subsets of $\{1,\ldots,n\}$ and in which two vertices are adjacent if one is a subset of the other. This graph is a supergraph of the $n$-dimensional hypercube and we note some similarities between our results and Harper's theorem, which solves the isoperimetric problem for hypercubes. In particular, analogously to Harper's theorem, we show there is a total ordering of the subsets 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 Gerbner et al. on cross-Sperner families and allow us to strengthen one of their main results.