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**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 I_{1},I_{2} be an ordered pair of sets and G: I_{1} ––> I_{2} a mapping of I_{1} into I_{2}; G is called an inner set mapping provided G X \subset X for every X in I_{1}. The inverse of any X_{0} in I_{2} is defined in two ways: as X_{0}^{-1} = \bigcup X (GX = X_{0}) and as X_{0}^{*-1} = **{**X; G(X) = X_{0} **}**. 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 X_{0} in [S]^{p} satisfying card X_{0}^{-1} = r and card X_{0}^{*-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 \aleph_{0} ==> ((m,q,q)) (not)––> q^+.

Theorem 5: q \geq \aleph_{0} ==> ((m,q,q))^* (not)––> 2. Theorem 6: p < q, q^{p} < m^{q}, q \geq \aleph_{0}, q^{p} < m^* ==> ((m,p,q)) ––> m. If moreover the general continuum hypothesis is assumed and q^{p} \ne m^* or q \geq m^*, then ((m,p,q))^* ––> m^{q} and ((m,p,q)) ––> m (Theorem 9). Let \alpha be an ordinal; if 0 < k < l < \aleph_{0}, 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 r^{n} < m for every r < q and n < q then ((m, < q,q)) ––> m (Theorem 12). Problems: Does subsist ((\aleph_{\omega_{\omega+1}}, \aleph_{0}, \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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