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**Zentralblatt MATH**

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

**Zbl.No: ** 277.04006

**Autor: ** Erdös, Paul; Hajnal, András; Milner, E.C.

**Title: ** Partition relations for \eta_{\alpha} and for \aleph_{\alpha}-saturated models. (In English)

**Source: ** Theory Sets Topology, Collection Papers Honour Felix Hausdorff, 95-108 (1972).

**Review: ** [For the entire collection see Zbl 256.00006.]

Let R_{\alpha} be the set of all non-zero dyadic sequences (x_{\nu}; \nu < \omega_{a}lpha) which are eventually zero, this is, there is some \nu < \omega_{\alpha} such that x_{\nu} = 1 and x_{\mu} = 0 for \mu > \nu. The order type of R_{\alpha} under the lexicographic ordering is denoted by \eta_{\alpha}. The following two theorems are proved (under the assumption of the Generalized Continuum Hypothesis): I. if \aleph_{\alpha} is regular and m is a cardinal with m < \aleph_{a}lpha then \eta_{\alpha} ––> (\eta_{\alpha} , [m, \eta_{\alpha}])^{2}. For all \beta, \eta_{\beta+1} ––> (\eta_{\beta+1}[\eta_{\beta},\eta_{\beta}])^{2}.

The symbol \eta_{\alpha} ––> (\eta_{\alpha},[m,\eta_{\alpha}])^{2} means the following: Whenever the set of unordered pairs of elements from R_{\alpha} is partitioned into two classes either there is a subset X of R_{\alpha} of order type \eta_{\alpha} all the pairs from which lie in the first class, or else there are subsets M, N of R_{\alpha} where M has power m and N has order type \eta_{\alpha} such that all the pairs **{**x,y**}** where x in M, y in N, x \ne y fall into the second class. The second partition symbol has an analogous meaning.

In fact, results more general than these are established. These are applied to show, in particular, that results corresponding to I and II hold if the set R_{\alpha} of order type \eta_{\alpha} is replaced by certain \aleph_{\alpha}-saturated models. Other partition relations for \eta_{\alpha}-sets have appeared in an earlier paper of *P. Erdös, E. C. Milner* and *R. Rado* [J. London math. Soc., II. Ser. 3, 193-204 (1971; Zbl 212.02204)].

**Reviewer: ** N.H.Williams

**Classif.: ** * 04A20 Combinatorial set theory

03C99 Model theory (logic)

04A10 Ordinal and cardinal numbers; generalizations

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