Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 174.01804
Autor: Erdös, Pál; Hajnal, András; Milner, E.C.
Title: On sets of almost disjoint subsets of a set (In English)
Source: Acta Math. Acad. Sci. Hung. 19, 209-218 (1968).
Review: 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 \aleph0 and if n < m implies 2n < m, then there is no maximal set of power m of almost disjoint transversals of \aleph0 disjoint sets of power \aleph0.
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
Classif.: * 05D15 Transversal (matching) theory
04A20 Combinatorial set theory
Index Words: set theory
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