On a biased edge isoperimetric inequality for the discrete cube

David Ellis, Nathan Keller, Noam Lifshitz

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

The ‘full’ edge isoperimetric inequality for the discrete cube {0,1}n (due to Harper, Lindsey, Berstein and Hart) specifies the minimum size of the edge boundary ∂A of a set A⊂{0,1}n, as function of |A|. A weaker (but more widely-used) lower bound is |∂A|≥|A|log⁡(2n/|A|), where equality holds whenever A is a subcube. In 2011, the first author obtained a sharp ‘stability’ version of the latter result, proving that if |∂A|≤|A|(log⁡(2n/|A|)+ϵ), then there exists a subcube C such that |AΔC|/|A|=O(ϵ/log⁡(1/ϵ)). The ‘weak’ version of the edge isoperimetric inequality has the following well-known generalization for the ‘p-biased’ measure μp on the discrete cube: if p≤1/2, or if 0<p<1 and A is monotone increasing, then pμp(∂A)≥μp(A)logp⁡(μp(A)). In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if pμp(∂A)≤μp(A)(logp⁡(μp(A))+ϵ), then there exists a subcube C such that μp(AΔC)/μp(A)=O(ϵ/log⁡(1/ϵ)), where ϵ:=ϵln⁡(1/p). This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős–Ko–Rado type theorems in extremal combinatorics, including the seminal ‘complete intersection theorem’ of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the ‘full’ edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the ‘full’ edge isoperimetric inequality, one relying on the Kruskal–Katona theorem.

Original languageAmerican English
Pages (from-to)118-162
Number of pages45
JournalJournal of Combinatorial Theory. Series A
Volume163
DOIs
StatePublished - Apr 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018 Elsevier Inc.

Keywords

  • Biased measure
  • Discrete cube
  • Isoperimetric inequalities
  • Stability

Fingerprint

Dive into the research topics of 'On a biased edge isoperimetric inequality for the discrete cube'. Together they form a unique fingerprint.

Cite this