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**Zentralblatt MATH**

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

**Zbl.No: ** 559.04009

**Autor: ** Baumgartner, James E.; Erdös, Paul; Higgs, D.

**Title: ** Cross-cuts in the power set of an infinite set. (In English)

**Source: ** Order 1, 139-145 (1984).

**Review: ** Authors' abstract: ``In the power set P(E) of a set E, the sets of a fixed finite cardinality k form a ``cross-cut'', that is, a maximal unordered set C such that if X,Y\subseteq E satisfy X\subseteq Y, X\subseteq some X' in C, and Y\supseteq some Y' in C, then X\subseteq Z\subseteq Y for some Z in C. For E = \omega, \omega_{1} and \omega_{2}, it is shown with the aid of the continuum hypothesis that P(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for \omega and \omega_{1}.''

**Reviewer: ** N.H.Williams

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

06A06 Partial order

04A30 Continuum hypothesis and generalizations

**Keywords: ** power set; sets of a fixed finite cardinality; cross-cut; maximal unordered set; continuum hypothesis

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