Ein Zerlegungssatz für P(\(\kappa\) )
\textit{P. Erdős} [Infinite and finite sets, Colloq. Math. Soc. Janos Bolyai 10, 403-424 (1975; Zbl 0361.05038)] asked: Is there a \(\kappa\geq \omega\) such that if \({\mathcal P}(\kappa)\) or \([\kappa]^{\omega}\) is divided into two parts then there are always disjoint \(A_ k\) for \(k\in \omega\) such that all of their (in)finite unions belong to the same part? The authors provide partial answers to this question. Some sample results: If \(\kappa\geq \omega\) and \(T:[\kappa]^{\kappa}\to cf(\kappa)\) is a partition, then there exists a \(\Delta\)-system \({\mathcal D}\subseteq [\kappa]^{\kappa}\) of size \(\kappa\) whose root is bounded in \(\kappa\), on which T is constant. 2. If \(\kappa,\lambda\geq \omega\) are such that \(2^{<\kappa}=\kappa\) and \(2^{\lambda}\leq (2^{\kappa})^{(+\omega)}\), then there is a \(T:{\mathcal P}(\lambda)\to 2^{\kappa}\) such that whenever \({\mathcal D}\subseteq {\mathcal P}(\lambda)\) is quasi-disjoint and of size \(\kappa\), there is for every \(\alpha <2^{\kappa}\) a \({\mathcal D}'\subseteq {\mathcal D}\) with T(\(\cup {\mathcal D}')=\alpha.\) Hence the \(\kappa\) wanted by Erdős must be bigger than the first limit above \(2^{\omega}\).
scientific article; zbMATH DE number 3878915