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On a classification of denumerable order types and an application to the partition calculus. (English) Zbl 0111.01201

Let \(\Theta\) and \(\Theta'\) be denumerable order types; \(\Theta\) is discrete if \(\eta \not\leq \Theta\). It is shown that if \(\Theta\) is discrete, \(\Theta\) has a rank \(\varrho (\Theta)\) (an ordinal \(< \omega_1\)) defined from the way \(\Theta\) is attainable from 0 and 1 via a transfinite process of \(\omega\)- and \(\omega^*\)-additions. It is shown that if \(\Theta\) is not discrete, \(\Theta\) is a sum of type \(\eta\), \(1+\eta\), \(\eta+1\), or \(1+\eta+1\) of non-zero discrete types. Among the theorems (here paraphrased) in the partition calculus proved by using rank are the following statements (the bracketed insertions have been made by the reviewer). \(\Theta \to (\Theta, \aleph_0)^2\) if (and only if) \(\Theta = \omega\) or \(\Theta = \omega^*\) or \(\eta \leq \Theta\) [or \(\Theta < 2\)]. \(\Theta \nrightarrow (\Theta',\aleph_0)^2\) if \(\Theta\) is discrete and \(\Theta'\neq n+\omega^*\) and \(\Theta' \neq \omega + n\) for each \(n < \omega\) [and \(\Theta'\) is infinite]. \(\Theta \to (\omega+n,\aleph_0)^2\) if and only if \(\omega \cdot \omega^* \leq \Theta\).
[Minor errors: On line 27 of p. 125 replace ”\(\overline{\overline{S'' \cdot S_{n'_0}}} = \aleph_0\)” by ”either both \(n_0<n_0'\) and \(\overline {\overline {S'' \cdot S_{n'_0}}} = \aleph\)”. Lines 18-20 of p. 125 neglect the possibility that \(\overline{\overline{S'}} = \aleph_0\) and \([S']^2 \subset I_2\); however, this possibility may be handled trivially.]
Reviewer: A.H.Kruse

MSC:

05D10 Ramsey theory
03E10 Ordinal and cardinal numbers
03E05 Other combinatorial set theory

Keywords:

set theory
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