Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  429.04005
Autor:  Erdös, Paul; Galvin, Fred; Rado, Richard
Title:  Transversals and multitransversals. (In English)
Source:  J. Lond. Math. Soc., II. Ser. 20, 387-395 (1979).
Review:  Let I be a set (of indices). Then, by definition, the disjoint subset relation (ai: i in I) ––> (bi: i in I)ds means that the ai, bi are cardinals with the property that whenever Ai is a set of cardinality ai there exist pairwise disjoint sets Xi in [Ai]bi (i in I). Families (Xi: i in I) are called multitransversals of (Ai: i in I) of size (bi: i in I). A transversal [multitransversal] of a family F of sets is a family of distinct elements [disjoint sets] one from each number of F [cf. also \S9, pp. 89-97 in reviewer's Thesis, Ensembles ordonnés et ramifiés, Paris (1935) and Publ. Math. Univ. Belgrade 4, 1-138 (1935; Zbl 014.39401)]. The authors consider 21 statements and prove the mutual equivalence of the statements (1), (2), (3), (4), (5) (Theorem 1), of statements (6), (7), (8), (9), (10) (Theorem 2), of statements (13), (14), (15), (16) (Theorem 3), respectively, and the following main results: ''
Theorem 4: Let I be a set ai, bi be arbitrary cardinals for i in I. Put S = {ai: i in I; bi \geq i}. Then (17) \leftrightarrow (18) \leftrightarrow (19) \land (20) \land (21), where (17) (ai: i in I) ––> (bi: i in I)ds, (18) (ai: i in I) has a multitransversal of size (bi: i in I); (19) \Sigma(i in I; ai \leq k)bi \leq k for every cardinal k; (20) \omega(S)\land \bar{\lambda}\notin stat\lambda for every weakly inaccessible cardinal \lambda; (21) if m < \omega and m \leq \Sigma(i in I; a1 = \aleph0)bi, then m+\Sigma(i in I; ai \leq n)bi \leq n for sufficiently large finite n. ''Notation: For a cardinal c, c denotes the set of all cardinals < c. For a regular cardinal \lambda, a set A is stationary on \bar{\lambda} if A\subset\bar{\lambda} and for every regressive function f on A there exists y < \lambda such that |f-1{y}| = \lambda, stat\lambda denotes the system of all sets which are stationary on \bar{\lambda}.
Reviewer:  \D.Kurepa
Classif.:  * 04A20 Combinatorial set theory
                   04A25 Axiom of choice and equivalent propositions
                   04A10 Ordinal and cardinal numbers; generalizations
                   03E10 Ordinal and cardinal arithmetic
Keywords:  transversal; stationary sets; disjoint subset relation; multitransversals; inaccessible cardinal
Citations:  Zbl.014.394

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