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 = (Bn: 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 = (Bn: n < \omega) is said to be a paradoxical decomposition of \xi if (i) tp. Bn < \kappan(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(\kappan)n < \omega1 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 \aleph2 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(\kappan)n < \omega1 if \xi < \kappa, which higher cardinal analogue is false if we assume 2\aleph1 = \aleph2.
Reviewer: P.L.Ferrari
Classif.: * 04A20 Combinatorial set theory
05A17 Partitions of integres (combinatorics)
03E55 Large cardinals
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