On set systems without a simplex-cluster and the junta method

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Abstract

A family {A0,…,Ad} of k-element subsets of [n]={1,2,…,n} is called a d-simplex-cluster if A0∩⋯∩Ad=∅, |A0∪⋯∪Ad|≤2k, and the intersection of any d of the sets in {A0,…,Ad} is nonempty. In 2010, Keevash and Mubayi conjectured that for any [Formula presented], the largest family of k-element subsets of [n] that does not contain a d-simplex-cluster is the family of all k-subsets that contain a given element. We prove the conjecture for all k≥ζn for an arbitrarily small ζ>0, provided that n≥n0(ζ,d). We call a family {A0,…,Ad} of k-element subsets of [n] a (d,k,s)-cluster if A0∩⋯∩Ad=∅ and |A0∪⋯∪Ad|≤s. We also show that for any [Formula presented] the largest family of k-element subsets of [n] that does not contain a [Formula presented]-cluster is again the family of all k-subsets that contain a given element, provided that n≥n0(ζ,d). Our proof is based on the junta method for extremal combinatorics initiated by Dinur and Friedgut and further developed by Ellis, Keller, and the author.

Original languageAmerican English
Article number105139
JournalJournal of Combinatorial Theory. Series A
Volume170
DOIs
StatePublished - Feb 2020

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Extremal combinatorics
  • Junta method
  • Set systems

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