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 \lambda1+k\ell ––> (\lambda1+k, \ell+1)2. On the other hand, \lambdak\ell (not)––> (\lambda1+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
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