Erdős, Paul; Hajnal, András Unsolved and solved problems in set theory. (English) Zbl 0334.04003 Proc. Tarski Symp., internat. Symp. Honor Alfred Tarski, Berkeley 1971, Proc. Symp. Pure Math. 25, 269-287 (1974). [For the entire collection see Zbl 0291.00009.] The problems in this paper are in the domain of combinatorial set theory. The major part of the paper is a report on solutions, or progress toward solutions, made by themselves and others, on problems in their previous paper [Axiomatic Set Theory, Proc. Sympos. pure Math. 13, Part I, 17-48 (1971; Zbl 0228.04001)]. In the last third of the paper, the authors pose and explain nine new problems or groups of problems, of which we list several below (chosen on the basis of brevity). Problem V (Erdős-Prikry). Let \(|S| = \aleph_1\), \([S]^{\aleph_1}= \cup_{\xi < \omega_1}I_\xi\). Does there exist \(\xi < \omega_1\) and sets \(A,B,C \in I_\xi\) such that \(A \cup B=C\)? Problem VI. Assume G.C.H. Let \(|S| = \aleph_2\). Does there exists a disjoint partition \([S]^2= \cup_{\nu < \omega_1}I_\nu\) satisfying the following condition: For all \(S' \subset S\), \(|S'| = \aleph_2\), there is \(Z \subset S\), \(|Z| = \aleph_1\) such that all different pairs of \(Z\) belong to different \(I_\nu\)? Problem IX. Assume G.C.H. and let \(\langle S,I \rangle\) establish \(\aleph_{\alpha +1} \nrightarrow ([\aleph_\alpha, \aleph_{\alpha +1}])^2_2\). (This means that \(|S| = \aleph_{\alpha +1}\), \([X]^2 \not\subset I\) for all \(x \in [S]^{\aleph_\alpha}\) and \([Y]^2 \cap I \neq \emptyset\) for all \(Y \in [S]^{\aleph_{\alpha +1}}\).) For what \(\beta \leq \alpha\) is \(\langle Z,g \rangle\) isomorphic to a substructure of \(\langle S,I \rangle\) for \(|Z| \leq \aleph_\beta\)? Reviewer: E.Mendelson Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 7 Documents MSC: 03E05 Other combinatorial set theory 00A07 Problem books Citations:Zbl 0291.00009; Zbl 0228.04001 PDFBibTeX XML