Well-Graded Families and the Union-Closed Sets Conjecture
نویسندگان
چکیده
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...
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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...
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We survey the state of the union-closed sets conjecture.
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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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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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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2020
ISSN: 1077-8926
DOI: 10.37236/8380