JournalsjemsVol. 21, No. 12pp. 3857–3902

Stability versions of Erdős–Ko–Rado type theorems via isoperimetry

  • David Ellis

    Queen Mary, University of London, UK
  • Nathan Keller

    Bar-Ilan University, Ramat Gan, Israel
  • Noam Lifshitz

    The Hebrew University of Jerusalem, Israel
Stability versions of Erdős–Ko–Rado type theorems via isoperimetry cover
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Abstract

Erdős–Ko–Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family.

In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results.

We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede–Khachatrian theorem on tt-intersecting families of kk-element subsets of {1,,n}\{1,\dots,n\} (for k<n/(t+1)k < n/(t+1)), and to show that, somewhat surprisingly, all these results hold when the ‘intersection’ requirement is replaced by a much weaker requirement.

Other examples include stability versions of Frankl’s recent result on the Erdős matching conjecture, the Ellis–Filmus–Friedgut proof of the Simonovits–Sós conjecture, and various EKR-type results on rr-wise (cross-)tt-intersecting families.

Cite this article

David Ellis, Nathan Keller, Noam Lifshitz, Stability versions of Erdős–Ko–Rado type theorems via isoperimetry. J. Eur. Math. Soc. 21 (2019), no. 12, pp. 3857–3902

DOI 10.4171/JEMS/915