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**Zentralblatt MATH**

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

**Zbl.No: ** 378.04002

**Autor: ** Erdös, Paul; Hajnal, András; Milner, E.C.

**Title: ** On set systems having paradoxical covering properties. (In English)

**Source: ** Acta Math. Acad. Sci. Hung. 31, 89-124 (1978).

**Review: ** Let \xi be an ordinal, \kappa a cardinal so that \xi < \kappa^+. A family B = (B_{n}: n < \omega) of subsets of \xi is said to have the \omega-covering property if the union of any \omega of these sets is the whole set \xi . On the other hand, the family B = (B_{n}: n < \omega) is said to be a paradoxical decomposition of \xi if (i) tp. B_{n} < \kappa^{n}(n < \omega) and (ii) B has the \omega-covering property. An example of paradoxical decomposition is given from the theorem of Milner and Rado \xi\twoheadrightarrow(\kappa^{n})_{n < \omega}^{1} if \xi < \kappa^+ The existence of such a partition is related with some results in the theory of polarized partition relations (the authors in Studies pure Math.,63-87 (1971; Zbl 228.04002)). This paper contains a study of \aleph_{2} phenomena, i.e. of such partition relations whose ``next higher case'' (i.e. the formula obtained by replacing each cardinal by its successor) is not true. The main reason why it is not possible to extend in a symple way such results is that one of principal tools which were used was the Milner-Rado paradoxical decomposition \xi\twoheadrightarrow(\kappa^{n})_{n < \omega}^{1} if \xi < \kappa, which higher cardinal analogue is false if we assume 2^{\aleph1} = \aleph_{2}.

**Reviewer: ** P.L.Ferrari

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

05A17 Partitions of integres (combinatorics)

03E55 Large cardinals

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