On union-closed families

"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

Union-closed families of sets

Abstract3e use a lower bound on the number of small sets in an ideal to show that for each union-closed family of n sets there exists an element which belongs to at least of them, provided n is large enough

Testing Intersecting and Union-Closed Families

Inspired by the classic problem of Boolean function monotonicity testing, we investigate the testability of other well-studied properties of combinatorial finite set systems, specifically \emph{intersecting} families and \emph{union-closed} families. A function $f: \{0,1\}^n \to \{0,1\}$ is intersecting (respectively, union-closed) if its set of satisfying assignments corresponds to an intersecting family (respectively, a union-closed family) of subsets of $[n]$. Our main results are that -- in sharp contrast with the property of being a monotone set system -- the property of being an intersecting set system, and the property of being a union-closed set system, both turn out to be information-theoretically difficult to test. We show that: $\bullet$ For $\epsilon \geq \Omega(1/\sqrt{n})$, any non-adaptive two-sided $\epsilon$-tester for intersectingness must make $2^{\Omega(n^{1/4}/\sqrt{\epsilon})}$ queries. We also give a $2^{\Omega(\sqrt{n \log(1/\epsilon)})}$-query lower bound for non-adaptive one-sided $\epsilon$-testers for intersectingness. $\bullet$ For $\epsilon \geq 1/2^{\Omega(n^{0.49})}$, any non-adaptive two-sided $\epsilon$-tester for union-closedness must make $n^{\Omega(\log(1/\epsilon))}$ queries. Thus, neither intersectingness nor union-closedness shares the $\mathrm{poly}(n,1/\epsilon)$-query non-adaptive testability that is enjoyed by monotonicity. To complement our lower bounds, we also give a simple $\mathrm{poly}(n^{\sqrt{n\log(1/\epsilon)}},1/\epsilon)$-query, one-sided, non-adaptive algorithm for $\epsilon$-testing each of these properties (intersectingness and union-closedness). We thus achieve nearly tight upper and lower bounds for two-sided testing of intersectingness when $\epsilon = \Theta(1/\sqrt{n})$, and for one-sided testing of intersectingness when $\epsilon=\Theta(1).$Comment: To appear in ITCS'2