Union-closed families of sets

Balla, Igor; Bollobás, Béla; Eccles, Tom

doi:10.1016/j.jcta.2012.10.005

Cited by 25 publications

(44 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we recall some concepts used by Reimer [8] and Balla, Bollobás and Eccles [1] in their work on the union-closed size problem. Central to both of those papers are compressions.…”

Section: Definitionsmentioning

confidence: 99%

“…Following the approach of [1], we shall view the complement of a union-closed family as a simply rooted family -this perspective is crucial for our proof of Theorem 1.3. We call a family B ⊆ P(n) simply rooted if for every…”

Section: Definitionsmentioning

confidence: 99%

“…This is a linear order on N (<∞) . We write I(m) for the initial segment of this order of length m; so, for example, I(9) = {∅, 1,2,12,3,13,23,123, 4}, where we write 13 for the set {1, 3}. Also, a family of sets D is called a down-set if for every A ∈ D we have P(A) ⊆ D. The following result is a well-known consequence of the fundamental theorem of Kruskal [7] and Katona [6].…”

Section: Definitionsmentioning

confidence: 99%

“…The other fact which we shall need about initial segments of colex is the following lemma, which is a simple corollary of Lemma 2.2 -see for example [1]. Lemma 2.3.…”

Section: Definitionsmentioning

confidence: 99%

“…The extremal family A for the first part of the theorem has P(n)\A = {B∪{n} : B ∈ I(m ′ )}. Through bounding ||I(m ′ )||, this result is sufficient to prove the union-closed conjecture if |A| is large -in fact the following bound is given in [1]. Corollary 1.2.…”

Section: Introductionmentioning

confidence: 98%

See 4 more Smart Citations

A Stability Result for the Union-Closed Size Problem

Eccles¹

2015

Combinator. Probab. Comp.

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Definitionsmentioning

confidence: 99%

Section: Definitionsmentioning

confidence: 99%

Section: Definitionsmentioning

confidence: 99%

“…The other fact which we shall need about initial segments of colex is the following lemma, which is a simple corollary of Lemma 2.2 -see for example [1]. Lemma 2.3.…”

Section: Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 3 more Smart Citations