Uniqueness in Harper's vertex-isoperimetric theorem

TL;DR

This paper investigates the uniqueness of subsets in the hypercube that minimize neighborhood sizes, providing counterexamples to a conjecture and classifying all such extremal sets.

Contribution

It offers the first counterexample to a conjecture about minimal neighborhoods, classifies all extremal sets, and links minimal neighborhood sizes for different t.

Findings

Counterexamples to the conjecture are provided.

Sets with minimal neighborhoods are classified.

Minimal neighborhood properties for A and A^c are equivalent.

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.