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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 162.02001

**Autor: ** Erdös, Pál; Ulam, S.

**Title: ** On equations with sets as unknowns (In English)

**Source: ** Proc. Natl. Acad. Sci. USA 60, 1189-1195 (1968).

**Review: ** The authors prove among others the following theorems: Let |S| \geq \aleph_{0}, 2 \leq k_{1} \leq k_{2} \leq ...,k_{n} ––> oo and S = \bigcup^{kn}_{l = 1} A_{l}^{(n)}, n = 1,2,... be a decomposition of S into k_{n} disjoint sets. Then there is always an l_{n}, 1 \leq l_{n} \leq k_{n} so that **|**S - \bigcup^{oo}_{n = l} A_{ln}^{(n)} **|** \geq \aleph_{0}. Assume 2^{\aleph0} = \aleph_{1}, |S| = \aleph_{1}. Then S can be decomposed in \aleph_{1} ways as the union of disjoint sets S = \bigcup_{1 \leq \alpha < \omega1} A_{\alpha}^{(\beta)}, 1 \leq \beta < \omega_{1} so that if we choose any one of the sets A_{\alpha1}^{(\beta1)} for \aleph_{0} different \beta_{1} then **|**S-\bigcup^{oo}_{l = 1} A_{\alpha1}^{(\beta1)} **|** \leq \aleph_{0}. Several extensions and generalizations are discussed and many unsolved problems and relations with other problems and results in set theory are discussed.

**Classif.: ** * 05D05 Extremal set theory

04A20 Combinatorial set theory

**Index Words: ** set theory

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