An Attempt at Frankl’s Conjecture
نویسنده
چکیده
In 1979 Frankl conjectured that in a finite union-closed family F of finite sets, F = {∅} there has to be an element that belongs to at least half of the sets in F . We prove this when | F| 10.
منابع مشابه
Frankl’s Conjecture for Large Semimodular and Planar Semimodular Lattices
A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let m denote...
متن کاملRemarks on Frankl's conjecture
First a few reformulations of Frankl’s conjecture are given, in terms of reduced families or matrices, or analogously in terms of lattices. These lead naturally to a stronger conjecture with a neat formulation which might be easier to attack than Frankl’s. To this end we prove an inequality which might help in proving the stronger conjecture.
متن کاملOn the Scope of Averaging for Frankl's Conjecture
Let F be a union-closed family of subsets of an m-element set A. Let n = |F| ≥ 2. For b ∈ A let w(b) denote the number of sets in F containing b minus the number of sets in F not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over ...
متن کاملThe union-closed sets conjecture almost holds for almost all random bipartite graphs
Frankl’s union-closed sets conjecture states that in every finite unionclosed family of sets, not all empty, there is an element in the ground set contained in at least half of the sets. The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets. We prove...
متن کاملFormalizing Frankl's Conjecture: FC-Families
The Frankl’s conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. FC-families are families for which it is proved that every union-closed family containing them satisfies the Frankl’s condition (e.g., in every union-closed family that contains a oneelement set a, the element a is contain...
متن کامل