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Extensions of the Erdös-Rado theorem. (English) Zbl 0846.03021

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, 1-17 (1993).
We find here some extensions of the Erdös-Rado Theorem that answer some longstanding problems. Ordinary partition relations for cardinal numbers are fairly well understood [see P. Erdös, A. Hajnal, A. Máté and R. Rado, Combinatorial set theory: Partition relations for cardinals (1984; Zbl 0573.03019)], but for ordinal numbers much has been open, and much remains open. For example, any proof of the simplest version of the Erdös-Rado Theorem seems to yield: \[ \text{For any regular cardinal} \quad \kappa, \quad \text{if} \quad \mu < \kappa \quad \text{then} \quad (2^{< \kappa})^+ \to (\kappa + 1)^2_\mu, \] but to replace \(\kappa + 1\) by \(\kappa + 2\) seems quite a nontrivial problem. In this paper we will prove that if \(\kappa\) is regular and uncountable, then:
Theorem 3.1. \(\forall k < \omega\) \(\forall \xi < \log \kappa\) \((2^{< \kappa})^+ \to (\kappa + \xi)^2_k\).
Theorem 4.1. \(\forall n,k < \omega\) \((2^{< \kappa})^+ \to (\rho, (\kappa + n)_k)^2\), where \(\rho = \kappa^{\omega + 2} + 1\).
Theorem 5.1. \(\forall n < \omega\) \((2^{< \kappa})^+ \cdot \omega \to (\kappa \cdot n)^2_2\).
The actual version of 5.1 is slightly stronger. Here \(\log \kappa\) is the least cardinal \(\mu\) such that \(2^\mu \geq \kappa\), the exponentiation in 4.1 is ordinal exponentiation, and the products in 5.1 represent ordinal multiplication. For \(\kappa = \omega\), 3.1, 4.1, and 5.1 all follow from the known result \(\omega_1 \to (\alpha)^2_k\) for all \(\alpha < \omega_1\) and all \(k < \omega\), so the uncountable case is the interesting one.
For the entire collection see [Zbl 0780.00039].

MSC:

03E05 Other combinatorial set theory

Citations:

Zbl 0573.03019
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