A Stability Result for the Union-Closed Size Problem

Abstract

A family of sets is called union-closed if whenever A and B are sets of the family, so is A ∪ B. The long-standing union-closed conjecture states that if a family of subsets of [n] is union-closed, some element appears in at least half the sets of the family. A natural weakening is that the union-closed conjecture holds for large families; that is, families consisting of at least p02 n sets for some constant p0. The first result in this direction appears in a recent paper of Balla, Bollobás and Eccles [1], who showed that union-closed families of at least 2 3 2 n sets satisfy the conjecturethey proved this by determining the minimum possible average size of a set in a union-closed family of given size. However, the methods used in that paper cannot prove a better constant than 2 3 . Here, we provide a stability result for the main theorem of [1], and as a consequence we prove the union-closed conjecture for families of at least ( 2 3 − c)2 n sets, for a positive constant c.A related problem is the union-closed size problem, which asks how small the sets of a union-closed family can be. For a finite family A ⊆ P(n), we define the total size of A to be ||A|| = A∈A |A|.Then the union-closed size problem asks what is the value ofwhere the minimum runs over union-closed families which consist of m sets. This problem was first addressed by Reimer [8] in 2003, who proved thatRecently, Balla, Bollobás and Eccles [1] settled the union-closed size problem entirely, determining the exact value of f (m) for all m. We denote by I(m) the initial segment of the colex order on N (<∞) of length m; this order shall be defined fully in Section 2.Theorem 1.1. Let m be a positive integer, and let n be the unique integer with 2 n−1 < m ≤ 2 n . Set m ′ = 2 n − m. Then

Well-Graded Families and the Union-Closed Sets Conjecture

“…We say that x is abundant in F if d(x) 1 2 |F|. The union-closed sets conjecture, originally attributed to P. Frankl [20], states that if F ⊆ P(n) is union-closed, then there must be some x ∈ [n] that is contained in at least half of the sets of F; in other words, there is at least one element in [n] that is abundant in F. Some of the more recent examples of work related to the conjecture are given by [1,8,13,17,18], and for a thorough survey of the various results pertaining to the conjecture, as well as an introduction to many of the techniques used in these results, see [4]. In this work we will explore the connection between the union-closed sets conjecture and union-closed families that have the property of being well-graded.…”

New Conjectures for Union-Closed Families

“…The union-closed conjecture holds for any family F where m(F ) ≥ 2 3 2 n(F ) . Even more recently, Eccles strengthened this result by proving a stability version in [Ecc15]. Secondly, instead of proving the Frankl conjecture, one could instead try to prove that any union-closed family contains an element present in at least some fraction of the sets, just as Knill did in the following theorem.…”

Two Results on Union-Closed Families

“…Czédli [5] proved that for any union-closed family F ⊂ 2 [n] , where |F | ≥ 2 n − 2 n/2 , the conjecture holds. This was significantly improved by Balla, Bóllobas and Eccles [2] to all union-closed families of subsets of [n] of size at least 2 3 2 n , and then further improved by Eccles [6] to ( 2 3 − 1 104 )2 n . Our first theorem, is that the bound can be improved to 1 2 2 n : Theorem 1.2.…”