✦ LIBER ✦

On Union-Closed Families, I

✍ Scribed by Robert T. Johnson; Theresa P. Vaughan

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 174 KB
Volume: 84
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.1998.2888

No coin nor oath required. For personal study only.

✦ Synopsis

A union closed family A is a finite family of sets such that the union of any two sets in A is also in A. The conjecture under consideration is Conjecture 1: For every union closed family A, there exists some x contained in at least half the members of A. We study the structure of such families (as partially ordered sets), and verify the conjecture for a large number of cases.

1998 Academic Press

1. PRELIMINARIES

A union closed (UC) family A is a finite family of sets, not all empty, such that the union of any two sets in A is also in A. The conjecture under consideration was proposed by Peter Frankl in 1979: Conjecture 1. For every UC family A, there exists some x contained in at least half the members of A.

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