Elekes, G.; Erdős, Paul; Hajnal, András On some partition properties of families of sets. (English) Zbl 0474.04002 Stud. Sci. Math. Hung. 13, 151-155 (1978). 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 Document MSC: 03E05 Other combinatorial set theory Keywords:partitions of the power set of an infinite cardinal; homogeneous sets PDFBibTeX XMLCite \textit{G. Elekes} et al., Stud. Sci. Math. Hung. 13, 151--155 (1978; Zbl 0474.04002)