Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 094.03204
Autor: Erdös, Pál; Fodor, G.; Hajnal, András
Title: On the structure of inner set mappings. (In English)
Source: Acta Sci. Math. 20, 81-90 (1959).
Review: Let I1,I2 be an ordered pair of sets and G: I1 ––> I2 a mapping of I1 into I2; G is called an inner set mapping provided G X \subset X for every X in I1. The inverse of any X0 in I2 is defined in two ways: as X0-1 = \bigcup X (GX = X0) and as X0*-1 = {X; G(X) = X0 }. For any set S and any cardinal n let [S]n and [S] < n denote the family of all subsets of S, each of the cardinality n and < n respectively. The mappings on (resp. into) [S]n, [S] < n are called of type (resp. of range) n and < n respectively. For any cardinal n let n^* be the smallest cardinal such that n be the sum of n^* cardinals < n. In connection with set mappings of type q, and of range p of subsets of S,S being of cardinality m, let ((m,p,q)) ––> r and ((m,p,q))^* ––> r respectively mean that for every set mapping of [S]q into [S]p there exists an X0 in [S]p satisfying card X0-1 = r and card X0*-1 = r respectively. Analogously the authors define ((m, < p,q))^* ––> r. Twelve theorems and several problems concerning the foregoing notions are proved and formulated respectively; here are some ones.
Theorem 3: q \geq \aleph0 ==> ((m,q,q)) (not)––> q^+.
Theorem 5: q \geq \aleph0 ==> ((m,q,q))^* (not)––> 2. Theorem 6: p < q, qp < mq, q \geq \aleph0, qp < m^* ==> ((m,p,q)) ––> m. If moreover the general continuum hypothesis is assumed and qp \ne m^* or q \geq m^*, then ((m,p,q))^* ––> mq and ((m,p,q)) ––> m (Theorem 9). Let \alpha be an ordinal; if 0 < k < l < \aleph0, then ((\aleph\alpha+k,k,l)) ––> \aleph\alpha (Theorem 10) but ((\aleph\alpha+k,k,l)) (not)––> \aleph\alpha+1 (Theorem 11). If q is infinite and regular and rn < m for every r < q and n < q then ((m, < q,q)) ––> m (Theorem 12). Problems: Does subsist ((\aleph\omega_{\omega+1}, \aleph0, \aleph\omega)) ––> \aleph\omega_{\omega+1}? If n > \aleph\omega, does ((m, < \aleph\omega,\aleph\omega)) ––> n hold for some m?
Reviewer: G.Kurepa
Classif.: * 05D10 Ramsey theory
04A20 Combinatorial set theory
03E05 Combinatorial set theory (logic)
Index Words: set theory
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