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**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 \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

**Classif.: ** * 05D15 Transversal (matching) theory

04A20 Combinatorial set theory

**Index Words: ** set theory

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag