## 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 \aleph0, 2 \leq k1 \leq k2 \leq ...,kn ––> oo and S = \bigcupknl = 1 Al(n), n = 1,2,... be a decomposition of S into kn disjoint sets. Then there is always an ln, 1 \leq ln \leq kn so that |S - \bigcupoon = l Aln(n) | \geq \aleph0. Assume 2\aleph0 = \aleph1, |S| = \aleph1. Then S can be decomposed in \aleph1 ways as the union of disjoint sets S = \bigcup1 \leq \alpha < \omega1 A\alpha(\beta), 1 \leq \beta < \omega1 so that if we choose any one of the sets A\alpha1(\beta1) for \aleph0 different \beta1 then |S-\bigcupool = 1 A\alpha1(\beta1) | \leq \aleph0. 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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