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 < \omegaalpha) 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 < \alephalpha 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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