Minimising the total number of subsets and supersets

Abstract

Let F be a family of subsets of a ground set {1,…,n} with |F|=m, and let F↕ denote the family of all subsets of {1,…,n} that are subsets or supersets of sets in F. Here we determine the minimum value that |F↕| 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,…,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,…,n} such that, for each initial segment F of this ordering, F↕ 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.