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**Zentralblatt MATH**

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

**Zbl.No: ** 474.04002

**Autor: ** Elekes, G.; Erdös, Paul; Hajnal, András

**Title: ** On some partition properties of families of sets. (In English)

**Source: ** Stud. Sci. Math. Hung. 13, 151-155 (1978).

**Review: ** The paper states (without detailed proofs) results and problems concerning the existence of certain types of homogeneous sets for partitions P(\kappa) = \cup**{**D_{\alpha}; \alpha < \mu**}** of the power set P(\kappa) of the infinite cardinal \kappa into \mu classes. We say H\subseteq P(\kappa) is homogeneous for the partition if there is some \alpha < \mu with H\subseteq D_{\alpha}. The first questions discussed concern homogeneous \Delta-systems. The family \Cal a is called a \lambda,\Delta-system if |a| = \lambda and A\cap B is the same for all distinct A, B from \Cal a. Results stated include: For any partition of P(\kappa) into \kappa classes and any cardinal \delta < \kappa, there is a homogeneous \lambda,\Delta-system. If \kappa is regular this holds for \lambda = \kappa as well. Further questions relate to homogeneous (\lambda,\mu)-systems. The family \Cal J is said to bee a (\lambda,\mu)-system if there is a family \Cal a with |a| = \lambda such that \Cal J is the collection of all non-empty unions of < \mu-size subfamilies of \Cal a these unions being different for different subfamilies. Typical results: For any \lambda < \kappa and any finite n, every partition of P(\kappa) into \kappa classes has a homogeneous \lambda,n-system. If \kappa is regular, this holds for \lambda = \kappa as well. If 2^{ < \kappa} = \kappa any such partition has a homogeneous \aleph_{0},\aleph_{0}-system, but 2^{\kappa} = \kappa^+ then there is such a partition with no \aleph_{1},\aleph_{0}-system.

**Reviewer: ** N.H.Williams

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

**Keywords: ** partitions of the power set of an infinite cardinal; homogeneous sets

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