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 language | American English |
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Article number | 105139 |
Journal | Journal of Combinatorial Theory - Series A |
Volume | 170 |
DOIs | |
State | Published - Feb 2020 |
Bibliographical note
Publisher Copyright:© 2019 Elsevier Inc.
Keywords
- Extremal combinatorics
- Junta method
- Set systems