Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  326.02050
Autor:  Erdös, Paul; Hechler, S.H.
Title:  On maximal almost-disjoint families over singular cardinals. (In English)
Source:  Infinite finite Sets, Colloq. Honour Paul Erdös, Keszthely 1973, Colloq. Math. Soc. Janos Bolyai 10, 597-604 (1975).
Review:  [For the entire collection see Zbl 293.00009.]
\kappa is infinite and |X| = \kappa. A collection F of subsets of X is a \kappa-maximal almost-disjoint family (\kappa-MADF) if (i) for y in F, |Y| = \kappa, (ii) if Y,Z in F then |Y \cap Z| < \kappa, (iii) if S \subset X and |S| = \kappa then there is a Y in F such that |Y \cap S| = \kappa. There are no \kappa-MADF's of cardinality \kappa if \kappa is regular. This paper concerns the case where \kappa is singular. Theorem (GCH). There exists a \kappa-MADF of cardinality \kappa iff \kappa is singular. Theorem. If \kappa is singular and cf(\kappa) = \lambda then it is consistent with ZFC that there exist \kappa-MADF's of very cardinality \mu \leq 2\lambda except \mu = \lambda. Open Problem. Is it consistent with ZFC that there exists a singular \kappa for which there are no \kappa-MADF's of cardinality \kappa? The paper contains several other theorems and problems.
Reviewer:  J.M.Plotkin
Classif.:  * 03E55 Large cardinals
                   03E35 Consistency and independence results (set theory)

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