Families of Sets with Intersecting Clusters

Abstract

A family of k-subsets A 1 , A 2 , . . . , A d on [n] = {1, 2, . . . , n} is called a (d, c)-cluster if the union A 1 ∪ A 2 ∪ · · · ∪ A d contains at most ck elements with c < d. Let F be a family of k-subsets of an n-element set. We show that for k ≥ 2 and n ≥ k + 2, if every (k, 2)-cluster of F is intersecting, then F contains no (k − 1)-dimensional simplices. This leads to an affirmative answer to Mubayi's conjecture for d = k based on Chvátal's simplex theorem. We also show that for any d satisfying 3 ≤ d ≤ k and n ≥ dk d−1 , if every (d, d+12 )-cluster is intersecting, then |F| ≤ n−1 k−1 with equality only when F is a complete star. This result is an extension of both Frankl's theorem and Mubayi's theorem.

Set systems with union and intersection constraints

“…Regarding exact results we have already observed that Conjecture 1 holds for d = 2 and d = 3. The only other known case for Conjecture 1 is when d = k, where it follows from an old result of Chvátal[2] (this was recently observed in[1]). E-mail address: mubayi@math.uic.edu (D. Mubayi) 1.…”

“…The only other known case for Conjecture 1 is when d = k, where it follows from an old result of Chvátal[2] (this was recently observed in[1]). E-mail address: mubayi@math.uic.edu (D. Mubayi) 1. Partially supported by NSF grant DMS 0653946.…”

“…Frankl and Füredi [4] showed that the answer is again n−1 k−1 as long as n is sufficiently large, and conjectured that this holds for all n 3k/2. The first author [8] recently proved their conjecture, and generalized it still further by introducing the concept of a d-cluster (the word d-cluster to describe this particular set of configurations is due to Chen, Liu and Wang [1] There has been further progress when one or more of the parameters is large. In [7], it is proved that Conjecture 1 holds for d = 4 and large n, while Keevash and the first author [6] recently proved Conjecture 1 in a different range of n, namely when k/n and n/2 − k are both bounded away from zero.…”

Unavoidable subhypergraphs: a-clusters

“…The case d = k follows from a theorem of Chvátal [5] as it was observed by Chen, Liu, and Wang [4]. Keevash and Mubayi [14] proved Conjecture 1 when both k/n and n/2 − k are bounded away from zero, and Mubayi and Ramadurai [18] for n > n 3 (k).…”

Invitation to intersection problems for finite sets

“…Chen, Liu and Wang [21] observed that the case k = d + 1 of Conjecture 5.2 is reduced to Conjecture 5.1, which is true by a result of Chvátal.…”