An isoperimetric inequality for the Hamming cube and some consequences

TL;DR

This paper establishes a new isoperimetric inequality for the Hamming cube, leading to stability results and implications for the number of maximal independent sets, with potential for broader applications.

Contribution

The paper introduces a novel isoperimetric inequality for the Hamming cube and explores its consequences, including stability results and bounds on maximal independent sets.

Findings

Proved an isoperimetric inequality involving a specific integral over the Hamming cube.

Derived stability results related to edge and vertex boundaries in the cube.

Estimated the asymptotic number of maximal independent sets in the Hamming cube.

Abstract

Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: \[ \int h_A^\beta d\mu \ge 2 \mu(A)(1-\mu(A)). \] Here $\mu$ is uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta=\log_2(3/2)$; and, for $S\subseteq V$ and $x\in V$, \[ h_S(x) = \begin{cases} d_{V \setminus S}(x) &\mbox{ if } x \in S, 0 &\mbox{ if } x \notin S \end{cases} \] (where $d_T(x)$ is the number of neighbors of $x$ in $T$). This implies inequalities involving mixtures of edge and vertex boundaries, with related stability results, and suggests some more general possibilities. One application, a stability result for the set of edges connecting two disjoint subsets of $V$ of size roughly $|V|/2$, is a key step in showing that the number of maximal independent sets in $Q_n$ is $(1+o(1))2n\exp_2[2^{n-2}]$. This asymptotic statement, whose proof will appear separately, was the original motivation for the present work.