We survey the state of the union-closed sets conjecture.

In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is union closed if it contains the union of any two sets in it. Let XF = ∪F∈FF . A union closed family of sets is separating if for any two distinct elements in F , th...

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead prove that 2a is an upper bound to the number of sets in a union-closed family on a ground set of n elements where each element is in at most a sets for all a, n...

Let F be a union-closed family of subsets of an m-element set A. Let n = |F| ≥ 2 and for a ∈ A let s(a) denote the number of sets in F that contain a. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a ∈ A with n − 2s(a) ≤ 0. Strengthening a result of Gao and Yu [7] we verify the conjecture for the particular case when m ≥ 3 and ...

A collection A of finite sets is closed under union if A, B ∈ A implies that A ∪ B ∈ A. The Union-Closed Sets Conjecture states that if A is a union-closed collection of sets, containing at least one non-empty set, then there is an element which belongs to at least half of the sets in A. We show that if q is the minimum cardinality of ∪A taken over all counterexamples A, then any counterexample...

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The ...

A family of sets is union-closed if it contains the union of any two of its elements. Some years ago, Reimer gave a lower bound for the average size of an element of a union-closed family consisting of m sets and, two years ago, Czédli did the same under the additional condition that our sets are contained in a set with n elements. Recently Tom Eccles and I have determined the minimum average s...

A family of sets A is said to be union-closed if {A∪B : A, B ∈ A} ⊂ A. Frankl’s conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the sets. Here we prove that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set, extending work of P...