Two Results on Union-Closed Families

Abstract

We show that there is some absolute constant c > 0, such that for any union-closed family F ⊆ 2 [n] , if |F| ≥ ( 1 2 − c)2 n , then there is some element i ∈ [n] that appears in at least half of the sets of F. We also show that for any union-closed family F ⊆ 2 [n] , the number of sets which are not in F that cover a set in F is at most 2 n−1 , and provide examples where the inequality is tight.

Small Sets in Union-Closed Families

“…If X is a set, a family F of subsets of X is said to be union-closed if the union of any two sets in F is also in F. The Union-Closed Conjecture (a conjecture of Frankl [5]) states that if X is a finite set and F is a union-closed family of subsets of X (with F = {∅}), then there exists an element x ∈ X such that x is contained in at least half of the sets in F. Despite the efforts of many researchers over the last forty-five years, and a recent Polymath project [7] aimed at resolving it, this conjecture remains wide open. It has only been proved under very strong constraints on the ground-set X or the family F; for example, Balla, Bollobás and Eccles [3] proved it in the case where |F| 2 3 2 |X| ; more recently, Karpas [6] proved it in the case where |F| ( 1 2 − c)2 |X| for a small absolute constant c > 0; and it is also known to hold whenever |X| 12 or |F| 50, from work of Vučković and Živković [11] and of Roberts and Simpson [9]. Note that the Union-Closed Conjecture is not even known to hold in the weaker form where we replace the fraction 1/2 by any other fixed > 0.…”

Small Sets in Union-Closed Families

“…If X is a set, a family F of subsets of X is said to be union-closed if the union of any two sets in F is also in F. The Union-Closed Conjecture (a conjecture of Frankl [5]) states that if X is a finite set and F is a union-closed family of subsets of X (with F = {∅}), then there exists an element x ∈ X such that x is contained in at least half of the sets in F. Despite the efforts of many researchers over the last forty-five years, and a recent Polymath project [7] aimed at resolving it, this conjecture remains wide open. It has only been proved under very strong constraints on the ground-set X or the family F; for example, Balla, Bollobás and Eccles [3] proved it in the case where |F| ≥ 2 3 2 |X| ; more recently, Karpas [6] proved it in the case where |F| ≥ ( 1 2 − c)2 |X| for a small absolute constant c > 0; and it is also known to hold whenever |X| ≤ 12 or |F| ≤ 50, from work of Vučković and Živković [11] and of Roberts and Simpson [9]. Note that the Union-Closed Conjecture is not even known to hold in the weaker form where we replace the fraction 1/2 by any other fixed ǫ > 0.…”

Notes on the Union Closed Sets Conjecture

“…For more details on the development of the conjecture and what is known about it see the survey [7]. In the recent years [1,2,6,12,18,21,28,29,30] have further been published investigating the conjecture with respect to one aspect or another. In this context, the collaborative effort in [16] should also be mentioned.…”