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**Zentralblatt MATH**

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

**Zbl.No: ** 846.03023

**Autor: ** Erdös, Paul; Hajnal, A.; Larson, Jean A.

**Title: ** Ordinal partition behavior of finite powers of cardinals. (In English)

**Source: ** Sauer, N. W. (ed.) et al., Finite and infinite combinatorics in sets and logic. Proceedings of the NATO Advanced Study Institute, Banff, Canada, April 21-May 4, 1991. Dordrecht: Kluwer Academic Publishers, NATO ASI Ser., Ser. C, Math. Phys. Sci. 411, 97-115 (1993).

**Review: ** In the notation of Erdös and Rado, the expression \alpha ––> (\beta, p)^{2} means that for any graph on \alpha either there is an independent subset of type \beta or there is a complete subgraph of size p. We discuss results for this relation where \alpha and \beta are both finite powers of some cardinal. In particular, assume that \lambda is either a regular cardinal or a strong limit cardinal and that k and \ell are positive integers. Then \lambda^{1+k\ell} ––> (\lambda^{1+k}, \ell+1)^{2}. On the other hand, \lambda^{k\ell} (not)––> (\lambda^{1+k}, 2^{\ell-1}+1)^{2} holds provided k \geq 4. We prove that the positive result is sharp if \lambda is a successor cardinal of the form \lambda = \theta^+= 2^{\theta}, while the negative result is sharp if the cofinality of \lambda is a weakly compact cardinal.

**Classif.: ** * 03E05 Combinatorial set theory (logic)

03E10 Ordinal and cardinal arithmetic

**Keywords: ** finite powers of cardinals; partition ordinals; graph; independent subset; complete subgraph; regular cardinal; strong limit cardinal; successor cardinal

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag