Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 111.01201
Autor: Erdös, Pál; Hajnal, András
Title: On a classification of denumerable order types and an application to the partition calculus (In English)
Source: Fundam. Math. 51, 117-129 (1962).
Review: 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 \rho (\Theta) (an ordinal < \omega1) 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 ––> (\Theta, \aleph0)2 if (and only if) \Theta = \omega or \Theta = \omega^* or \eta \leq \Theta [or \Theta < 2]. \Theta (not)––> (\Theta',\aleph0)2 if \Theta is discrete and \Theta'\ne n+\omega^* and \Theta' \ne \omega+n for each n < \omega [and \Theta' is infinite]. \Theta ––> (\omega+n,\aleph0)2 if and only if \omega · \omega^* \leq \Theta.
[Minor errors: On line 27 of p. 125 replace "\overline{\overline{S'' · Sn'0}} = \aleph0" by "either both n0 < n0' and \overline {\overline {S'' · Sn'0}} = \aleph". Lines 18-20 of p. 125 neglect the possibility that \overline{\overline{S'}} = \aleph0 and [S']2 \subset I2; however, this possibility may be handled trivially.]
Reviewer: A.H.Kruse
Classif.: * 05D10 Ramsey theory
04A10 Ordinal and cardinal numbers; generalizations
04A20 Combinatorial set theory
03E05 Combinatorial set theory (logic)
Index Words: set theory
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