On Erdős–Ko–Rado for random hypergraphs I

Abstract

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection.Denote by H k (n, p) the random family in which each k-subset of {1, . . . , n} is present with probability p, independent of other choices.

Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

“…While this work has been under review, there has been a vivid interest in questions related to random versions of the Erdős-Ko-Rado theorem (see [4,5,7,9,10,14,15]). In particular, as well as the results of Balogh, Bohman and Mubayi [3], the question concerning the structure of the largest intersecting family in the random setting has been addressed in [5,14,15] for various ranges of k and p. Moreover, an extension of the robust stability result for intersecting families, Lemma 2.3, has been considered in [9], implying that Theorem 1.3 can be extended to a larger range of k. We refer to these papers for further information.…”

“…The question of the range of p for which the largest intersecting family F ⊂ H k (n, p) is indeed the projection of a principal family has been successfully addressed in [3] for k < n 1/2−o (1) . For larger k, which we are mainly interested in, the problem seems to be more complicated, and has only been studied recently in [14], for constant p. We make no contribution to this question here. However, as well as the bounds on i(H k (n, p)), we are able to show stability for k = Θ(n) in the same range for p as in case (4) in Theorem 1.2.…”

On Erdős–Ko–Rado for Random Hypergraphs II

“…The original paper of Balogh et al dealt mostly with k < n 1/2−ε (for a fixed ε > 0). In a companion paper [14] we precisely settle the question for k up to about (1/4)n log n and suggest a possible general answer.…”

Sharp threshold for the Erdős-Ko-Rado theorem

“…This is a particularly difficult problem because the property of interest is not monotone; for small p and large p the property holds, but sometimes for some p in between, the property does not hold. Hamm and Kahn improved the results for n 1/3 ≪ k 1 2 √ n log n [14] and n = 2k + 1 [15]. Gauy, Hàn and Oliveira [13] extended [1] and gave the asymptotic size of largest intersecting family for all k and almost all p. Balogh, Das, Delcourt, Liu, and Sharifzadeh [3] gave additional results up to k n/4, but generally not as tight as [14].…”