On the union-closed sets conjecture
نویسندگان
چکیده
We give a new proof of theorem Lo Faro concerning minimal counterexamples to the union-closed conjecture, and we prove some results “ε-version” maximal frequency in families.
منابع مشابه
Note on the Union-Closed Sets Conjecture
The union-closed sets conjecture states that if a finite family of sets A 6 = {∅} is union-closed, then there is an element which belongs to at least half the sets in A. In 2001, D. Reimer showed that the average set size of a union-closed family, A, is at least 1 2 log2 |A|. In order to do so, he showed that all union-closed families satisfy a particular condition, which in turn implies the pr...
متن کاملA note on the union-closed sets conjecture
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...
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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 ...
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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 ...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2023
ISSN: ['1872-681X', '0012-365X']
DOI: https://doi.org/10.1016/j.disc.2023.113430