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On sets of almost disjoint subsets of a set. (English) Zbl 0174.01804

Two sets are said to be almost disjoint if the cardinality of their intersection is strictly less than the cardinality of either. A transversal of disjoint nonempty sets is a set contained in their union which has one element in common with each set. Sierpinski showed that \(m\) disjoint sets each of power \(m\) posses more than \(m\) almost disjoint transversals. In this paper a number of related results are presented. These include: 1. \(\aleph_{\alpha +1}\) disjoint sets of power \(\aleph_\alpha\) possess a maximal set of \(\aleph_{\alpha +1}\) almost disjoint transversals. (Every other transversals being not almost disjoint from one of them.) 2. \(\aleph_{\alpha +1}\) disjoint sets of power \(\aleph_\alpha\) possess a set of transversals whose intersection has cardinality strictly less than \(\aleph_\alpha\). 3. If the cofinality cardinal of \(m\) is \(\aleph_0\) and if \(n<m\) implies \(2^n<m\), then there is no maximal set of power \(m\) of almost disjoint transversals of \(\aleph_0\) disjoint sets of power \(\aleph_0\).
Several other related results, all concerned with maximality and cardinality of sets of transversals are also presented, as in a generalization of a result of F. C. Cater itself extending Sierpinski’s theorem.
Reviewer: D.Kleitman

MSC:

05D15 Transversal (matching) theory
03E05 Other combinatorial set theory

Keywords:

set theory
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References:

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