Set systems with union and intersection constraints

Abstract

Let 2 d k be fixed and n be sufficiently large. Suppose that G is a collection of k-element subsets of an n-element set, and |G| > n−1 k−1 . Then G contains d sets with union of size at most 2k and empty intersection. This extends the Erdős-Ko-Rado theorem and verifies a conjecture of the first author for large n.

On Mubayi's Conjecture and Conditionally Intersecting Sets

“…As described above, Mubayi [16] proved the conjecture for d = 3, and Mubayi [17] proved the conjecture for d = 4 when n is sufficiently large. Füredi andÖzkahya [11] and Mubayi and Ramadurai [18] independently improved this result by proving the conjecture for sufficiently large n, thus generalising the above-mentioned result by Frankl and Füredi [10]. Chen et al [4] proved Mubayi's Conjecture for d = k and Füredi andÖzkahya [11] proved that Mubayi's Conjecture even holds when d = k + 1.…”

“…For t = 1, this becomes the (d, s)-conditionally intersecting condition which, in turn for s = 2k, is the intersecting condition (2). More generally, the (d, s, t)-conditionally intersecting condition naturally generalises many previous intersecting conditions in the literature [2,5,6,10,9,18]; those are here given a useful common framework.…”

“…An indication that Conjecture 13 might be true is that the family F has size n−k−1 k−1 + 1 which, asymptotically, converges to n−1 k−1 , the size of star families on n elements which, in turn, are asymptotically the largest (d, 2k)-conditionally intersecting families; see [11,18].…”

Unavoidable subhypergraphs: a-clusters

“…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). The present authors also proved Conjecture 1 in 2007 for n > n 4 (k) with a different approach (unpublished).…”

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

“…Mubayi [22] proved his conjecture in the case where d = 3, k is fixed and n is sufficiently large. He has also showed a stability result for general fixed d. Specifically, he proved that if k, d are fixed and n tends to infinity, then any family F ⊆ [n] k that is free of (d, k, 2k)-cluster and whose size is n−1 k−1 (1 − o (1)), must satisfy |F \S| = o n−1 k−1 for some star S. In 2009, Mubayi and Ramadurai [23] applied Mubayi's stability result and proved that Conjecture 1.7 holds for any fixed k and d, provided that n is sufficiently large. In 2009, Füredi and Özkahya [14] gave a different proof of the result of Mubayi and Ramadurai and showed that if k and d are fixed and n is sufficiently large, then any F ⊆ [n] k whose size is greater than n−1 k−1 contains a special kind of a (d, k, 2k)-cluster.…”