Stability for vertex isoperimetry in the cube

TL;DR

This paper establishes a stability version of Harper's cube vertex isoperimetric inequality, showing near-minimal boundary sets are close to Hamming balls, with sharp estimates and applications to other combinatorial results.

Contribution

It provides a new stability theorem for the cube vertex isoperimetric inequality and extends similar stability results to the Kruskal--Katona Theorem and related combinatorial theorems.

Findings

Sets with near-minimal vertex boundary are close to Hamming balls.

Sharp estimates for vertex boundary based on distance from a ball.

Extensions to stability versions of other combinatorial theorems.

Abstract

We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum possible are close to (generalised) Hamming balls. Furthermore, we obtain a local stability result for ball-like sets that gives a sharp estimate for the vertex boundary in terms of the distance from a ball, and so our stability result is essentially tight (modulo a non-monotonicity phenomenon). We also give similar results for the Kruskal--Katona Theorem and applications to new stability versions of some other results in Extremal Combinatorics.