## 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(2^{n}/|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(2^{n}/|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 μ_{p} on the discrete cube: if p≤1/2, or if 0<p<1 and A is monotone increasing, then pμ_{p}(∂A)≥μ_{p}(A)log_{p}(μ_{p}(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 pμ_{p}(∂A)≤μ_{p}(A)(log_{p}(μ_{p}(A))+ϵ), then there exists a subcube C such that μ_{p}(AΔC)/μ_{p}(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 language | American English |
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Pages (from-to) | 118-162 |

Number of pages | 45 |

Journal | Journal of Combinatorial Theory - Series A |

Volume | 163 |

DOIs | |

State | Published - Apr 2019 |

Externally published | Yes |

### Bibliographical note

Funding Information:The research of N.K. was supported by the Israel Science Foundation (grant no. 402/13), the Binational US–Israel Science Foundation (grant no. 2014290), and by the Alon Fellowship.

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- Biased measure
- Discrete cube
- Isoperimetric inequalities
- Stability