On union-closed families

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Charles Frederick Renn (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
Theresa Vaughan

Abstract: "A union-closed family is a non-empty finite collection of non-empty sets that is closed under unions. Peter Frankl conjectured in 1979 that given a union-closed family, then there exists an element that occurs in at least half of the member sets. Despite its simplicity, the conjecture has defied any general proof. The result has been proved for families involving up to 9 elements. In this paper we attempt to approach the above conjecture from a broad perspective. We begin by imposing a numbering on all possible subsets, or collections, of the power set on n elements. From this, we investigate relationships between the numbering of a given collection and whether or not it is union-closed. We also look for a pattern in the distribution of union-closed families within the numbering for n < 6. For this work, we use a complete listing of the union-closed families. We make use of several custom computer applications written in C++ to produce complete union-closed family listings for n = 3, 4, 5."--Abstract from author supplied metadata.

Additional Information

Language: English
Date: 2007
union-closed family, non-empty, finite, collection, non-empty sets, unions, Peter Frankl
Set theory
Mathematical analysis

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