Publications of (and about) Paul Erdös
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 ni \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 Gi(i in I) each admitting an ni-transversal: a set Si with 1 \leq card (A \cap Si) < ni+1 for all A in Gi'', it is sufficient that either (i) \mu < \nu, or (ii) \mu = \nu+r (r finite) and sum (ni+1) \geq mr+m+2, or (iii) sum (ni+1) \geq \aleph0. 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 \aleph1 different cardinalities, any two having at most one common element, splits into \aleph0 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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