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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 304.04003

**Autor: ** Davies, R.O.; Erdös, Paul

**Title: ** Splitting almost-disjoint collections of sets into subcollections admitting almost-transversals. (In English)

**Source: ** Infinite finite Sets, Colloq. Honour Paul Erdös, Keszthely 1973, Colloq. Math. Soc. Janos Bolyai 10, 307-322 (1975).

**Review: ** [For the entire collection see Zbl 293.00009.]

Assuming AC, it is proved, given cardinals n_{i} \geq 1 for i in I \ne Ø, an integer m \geq 0, and ordinals \mu, \nu, that for the truth of the proposition ``every collection of \aleph_{\mu} sets of cardinal \aleph_{\nu}, any two having \leq m common elements, splits into subcollections G_{i}(i in I) each admitting an n_{i}-transversal: a set S_{i} with 1 \leq card (A \cap S_{i}) < n_{i}+1 for all A in G_{i}'', it is sufficient that either (i) \mu < \nu, or (ii) \mu = \nu+r (r finite) and **sum** (n_{i}+1) \geq mr+m+2, or (iii) **sum** (n_{i}+1) \geq \aleph_{0}. Some incomplete results are presented supporting the conjecture that the condition is also necessary (assuming GCH), as it is in the case when I is a singleton, due to *P. Erdös* and *A. Hajnal* [Acta Math. Acad. Sci. Hung. 12, 87-123 (1961; Zbl 201.32801)]. The authors do not know whether every collection of sets of \aleph_{1} different cardinalities, any two having at most one common element, splits into \aleph_{0} subcollections each admitting a 1-transversal.

**Classif.: ** * 04A20 Combinatorial set theory

04A25 Axiom of choice and equivalent propositions

04A30 Continuum hypothesis and generalizations

04A10 Ordinal and cardinal numbers; generalizations

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