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On some partition properties of families of sets. (English) Zbl 0474.04002

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 \(\mathcal a\) is called a \(\lambda,\Delta\)-system if \(|a|=\lambda\) and \(A\cap B\) is the same for all distinct \(A\), \(B\) from \(\mathcal a\). 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 \(\mathcal J\) is said to bee a \((\lambda,\mu)\)-system if there is a family \(\mathcal a\) with \(|a|=\lambda\) such that \(\mathcal J\) is the collection of all non-empty unions of \(<\mu\)-size subfamilies of \(\mathcal a\) 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

MSC:

03E05 Other combinatorial set theory
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