An approximate isoperimetric inequality for r-sets

TL;DR

This paper establishes a near-optimal approximate isoperimetric inequality for r-sets, providing bounds on the vertex boundary size relative to the set size and parameters n and r.

Contribution

It proves a new vertex-isoperimetric inequality for r-element subsets of an n-set, extending understanding of boundary sizes in combinatorial structures.

Findings

The boundary size is at least proportional to sqrt(n/(r(n-r))) times the set measure.

The inequality is sharp up to a constant factor for sets with measure bounded away from 0 and 1.

Provides a fundamental bound applicable to combinatorial and graph-theoretic problems.

Abstract

We prove a vertex-isoperimetric inequality for [n]^(r), the set of all r-element subsets of {1,2,...,n}, where x,y \in [n]^(r) are adjacent if |x \Delta y|=2. Namely, if \mathcal{A} \subset [n]^(r) with |\mathcal{A}|=\alpha {n \choose r}, then the vertex-boundary b(\mathcal{A}) satisfies |b(\mathcal{A})| \geq c\sqrt{\frac{n}{r(n-r)}} \alpha(1-\alpha) {n \choose r}, where c is a positive absolute constant. For \alpha bounded away from 0 and 1, this is sharp up to a constant factor (independent of n and r).