Structure and Supersaturation for Intersecting Families

Abstract

The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the minimum number of disjoint pairs that must appear in families larger than the extremal threshold. We study the minimum number of disjoint pairs in families of permutations and in kuniform set families, and determine the structure of the optimal families. Our main tool is a removal lemma for disjoint pairs. We also determine the typical structure of k-uniform set families without matchings of size s when n ≥ 2sk + 38s 4 , and show that almost all k-uniform intersecting families on vertex set [n] are trivial when n ≥ (2 + o(1))k.

On the structure of large sum-free sets of integers

“…Note that an example given by Frankl and Füredi [21] shows that this term cannot be reduced to below √ k. The problem of enumerating I 1 (n, k) was first investigated by Balogh, Das, Delcourt, Liu and Sharifzadeh [6]. Building on the of Balogh et al, Frankl and Kupavskii [22] and, independently, Balogh, Das, Liu, Sharifzadeh and Tran [7] established the asymptotic formula |I 1 (n, k)| = (n + o(1))2 ( n−1 k−1 ) for n ≥ 2k + 3 √ k ln k. Motivated by this result and the theorem of Ellis and Lifshitz, we make the following conjecture.…”

Sharp threshold for the Erdős–Ko–Rado theorem

“…Balogh, Das, Delcourt, Liu, and Sharifzadeh [3] also study the enumeration variant of the Erdős–Ko–Rado theorem, determining the order of magnitude of the log of the number of independent sets in $$$ K\left(n,k\right) $$$. Balogh, Das, Liu, Sharifzadeh, and Tran [4] and independently Kupavskii and Frankl [12] strengthened this to say that most independent sets of $$$ K\left(n,k\right) $$$ are contained in stars for $$$ n\ge 2k+c\sqrt{k\log k} $$$, where $$$ c $$$ is a large constant in [4] and $$$ c=2 $$$ in [12]. Balogh, Garcia, Li, and Wagner [5] push this $$$ c\sqrt{k\log k} $$$ down to $$$ 100\log k $$$.…”

Sharp threshold for the Erdős-Ko-Rado theorem

“…Balogh, Das, Delcourt, Liu, and Sharifzadeh [3] also study the enumeration variant of the Erdős-Ko-Rado theorem, determining the order of magnitude of the log of the number of independent sets in K(n, k). Balogh, Das, Liu, Sharifzadeh, and Tran [4] and independently Kupavskii and Frankl [11] strengthened this to say that most independent sets of K(n, k) are contained in stars for n 2k + c √ k log k, where c is a large constant in [4] and c = 2 in [11]. Balogh, Garcia, Li, and Wagner [5] push this c √ k log k down to 100 log k. They conjecture it is true down to n 2k + 2, since a simple computation shows that the families which show the tightness of the Hilton-Milner theorem [16] outnumber the trivial families for n = 2k + 1.…”