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

Abstract

A family {A 0 , . . . , A d } of k-element subsets of [n] = {1, 2, . . . , n} is called a simplex-cluster if A 0 ∩· · ·∩A d = ∅, |A 0 ∪· · ·∪A d | ≤ 2k, and the intersection of any d of the sets in {A 0 , . . . , A d } is nonempty. In 2006, Keevash and Mubayi conjectured that for any d + 1 ≤ k ≤ d d+1 n, the largest family of k-element subsets of [n] that does not contain a 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 ≥ n 0 (ζ, d).We call a family {A 0 , . . . , A d } of k-element subsets of [n] a (d, k, s)-cluster if A 0 ∩ · · · ∩ A d = ∅ and |A 0 ∪ · · · ∪ A d | ≤ s. We also show that for any ζn ≤ k ≤ d d+1 n the largest family of k-element subsets of [n] that does not contain a (d, k, ( d+1 d + ζ)k)-cluster is again the family of all k-subsets that contain a given element, provided that n ≥ n 0 (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.

On Mubayi's Conjecture and Conditionally Intersecting Sets

“…for n sufficiently larger than s − r, where r ′ is the smallest integer for which F r ′ = ∅. By the proof of Theorem 23, (15) is true when r ′ ≥ s − u, so suppose that r ′ < s − u (≤ k). For each integer r with s − u ≤ r ≤ k − 1, Theorem 24 implies that n s−r n r |F r | ≤ σ s−r (F r ) C .…”

“…However, Mubayi [17] provided a counterexample that showed that the conjecture could not be extended to values of d greater than or equal to 2 k . During the publication of this article, Lifshitz [15] also proved that Mubayi's Conjecture holds when 0 < ζn ≤ k ≤ d−1 d n and n is sufficiently larger than ζ and d, and that, under these conditions, the upper bound in Condition (2) can indeed be relaxed to ( d d−1 + ζ)k. The first main result of the present paper, Theorem 5 in Section 2, is a new partial verification of Mubayi's Conjecture. Namely, we prove that Mubayi's Conjecture holds for stable families F ⊆ [n] k of k-sets; these are the families that are invariant with respect to the shifting operation.…”

New results on simplex-clusters in set systems

“…This problem also had a long history (see [6,13,14,15,10] for some of the more significant developments) and was completely resolved recently in a paper of the author [3]. In 2010, Keevash and Mubayi extended both conjectures by hypothesizing that the same bound would hold for any F ⊆ [n] k containing no d-simplex-cluster, and very recently Lifshitz answered their question in the affirmative for all n > n 0 (d) in [12].…”

A note on hypergraphs without non-trivial intersecting subgraphs

“…Recently, Currier [4] proved this conjecture for all k ≥ d + 1 ≥ 3 and n ≥ 2k. The Chvátal Simplex Conjecture is still open in general for n < 2k and 3 ≤ d ≤ k − 2, and we refer the reader to [1,3,8,5,6,12,13,15] and their references for more results related to this conjecture.…”