منابع مشابه
Union-closed families of sets
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...
متن کاملRandom Union-Closed Families
We shall show that the union-closed families conjecture holds for a random union-closed family with high probability. This strengthens a recent result of Bruhn and Schaudt.
متن کاملMinimal Weight in Union-Closed Families
Let Ω be a finite set and let S ⊆ P(Ω) be a set system on Ω. For x ∈ Ω, we denote by dS(x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then there is some x ∈ Ω with dS(x) ≥ 12 |S|. We consider a related question. Define the weight of a family S to be w(S) := ∑ A∈S |A|. Suppose S is union-closed. How small can w(S) be? Reimer sho...
متن کاملNew Conjectures for Union-Closed Families
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...
متن کاملTwo Results on Union-Closed Families
We show that there is some absolute constant c > 0, such that for any union-closed family F ⊆ 2, if |F| ≥ ( 1 2 − c)2n, 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, the number of sets which are not in F that cover a set in F is at most 2, and provide examples where the inequality is tight.
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1999
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(98)00208-8