Intersecting families of discrete structures are typically trivial

Abstract

The study of intersecting structures is central to extremal combinatorics. A family of permutations F ⊂ S n is t-intersecting if any two permutations in F agree on some t indices, and is trivial if all permutations in F agree on the same t indices. A k-uniform hypergraph is t-intersecting if any two of its edges have t vertices in common, and trivial if all its edges share the same t vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for n sufficiently large with respect to t, the largest t-intersecting families in S n are the trivial ones. The classic Erdős-Ko-Rado theorem shows that the largest t-intersecting k-uniform hypergraphs are also trivial when n is large. We determine the typical structure of t-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollobás set-pairs inequality to bound the number of maximal intersecting families, which can then 225 be combined with known stability theorems. We also obtain similar results for vector spaces.

Structure and Supersaturation for Intersecting Families

“…Frankl [22] proved the conjecture in the range n ≥ (2s − 1)k − s + 1, showing that the extremal families can be covered by s − 1 elements. Adapting the methods of Balogh et al [3], we show that a slightly larger lower bound on n guarantees that almost all families without a matching of size s have a cover of size s − 1. The s = 2 case corresponds to intersecting families.…”

“…The s = 2 case corresponds to intersecting families. In this case, Balogh et al [3] showed that when n ≥ (3 + o(1))k, almost all intersecting families are trivial. Our final result improves the required bound on n to the asymptotically optimal n ≥ (2 + o(1))k. Indeed, when n = 2k, then the number of intersecting families is 3 1 2 ( n k ) = 3 ( n−1 k−1 ) , since we can freely choose at most one set from each complementary pair of k-sets {A, [n] \ A}.…”

“…Given two functions f and g of some underlining parameter n, if lim n→∞ f (n)/g(n) = 0, we write f = o(g). For a, b, c ∈ R + , we write 3…”

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.…”

On hypergraphs without loose cycles

“…There are very few results in the case r ą 2 and ex r pn, Hq " opn r q. The only known case is when H consists of two edges sharing t vertices [1,4]. Very recently, Mubayi and Wang [8] studied |Forb r pn, Hq| when H is a loose cycle.…”