Uniqueness in Harper's vertex-isoperimetric theorem

Abstract

For a set $A\subseteq Q_{n}=\left\{ 0,1\right\} ^{n}$ the $t$-neighbourhood of $A$ is $N^{t}\left(A\right)=\left\{ x\,:\,d\left(x,A\right)\leq t\right\}$, where $d$ denotes the usual graph distance on $Q_{n}$. Harper's vertex-isoperimetric theorem states that among the subsets $A\subseteq Q_{n}$ of given size, the size of the $t$-neighbourhood is minimised when $A$ is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if $A\subseteq Q_{n}$ is a subset of given size for which the sizes of both $N^{t}\left(A\right)$ and $N^{t}\left(A^{c}\right)$ are minimal for all $t>0$, does it follow that $A$ is isomorphic to an initial segment of the simplicial order? Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e.\ a set $A$ with $B\left(x,r\right)\subseteq A\subseteq B\left(x,r+1\right)$ for some $x\in Q_{n}$. We go further to classify all the sets $A\subseteq Q_{n}$ for which the sizes of both $N^{t}\left(A\right)$ and $N^{t}\left(A^{c}\right)$ are minimal for all $t>0$ among the subsets of $Q_{n}$ of given size. We also prove that, perhaps surprisingly, if $A\subseteq Q_{n}$ for which the sizes of $N\left(A\right)$ and $N\left(A^{c}\right)$ are minimal among the subsets of $Q_{n}$ of given size, then the sizes of both $N^{t}\left(A\right)$ and $N^{t}\left(A^{c}\right)$ are also minimal for all $t>0$ among the subsets of $Q_{n}$ of given size. Hence the same classification also holds when we only require $N\left(A\right)$ and $N\left(A^{c}\right)$ to have minimal size among the subsets $A\subseteq Q_{n}$ of given size.