Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 071.05105
Autor: Erdös, Pál; Rado, R.
Title: A partition calculus in set theory. (In English)
Source: Bull. Am. Math. Soc. 62, 427-489 (1956).
Review: The memoir is a natural sequel of some previous articles (Zbl 038.15301; Zbl 048.28203; Zbl 051.04003; Zbl 055.04903) initiated by F.P.Ramsey in 1930 [Proc. London math. Soc., II. Ser. 30, 264-286 (1930)], see also D.Kurepa, C. R. Soc. Sci. Varsovie, Cl. III 1939, 61-67; Acad. Sci. Slovenica, Ser. A 1953; Dissertationes IV/4, 67-92 (1953). For a set S and a cardinal r let [S]r = {X| X \subseteq S, |X| = r}; in particular [S]r = 0, provided |S| < r. The basic concept is the following relation: Given numbers a,k,r and a k-sequence b\nu (\nu < k); the relation a ––> [b0,b1,...,b\nu,...]rk is said to hold provided for a set S of cardinality a and for every partition of the set [S]r: Sr = \bigcupr < k K\nu there are a B\subseteq S and a \nu < k satisfying |B| = b\nu, [B]r \subseteq K. An analog relation is defined if a,b\nu be order types; in this case instead of |B| = b one considers the condition \bar B = b (\bar B denoting the order type of B). If b\nu is a constant sequence b0 the corresponding relation is denoted a ––> (b0)rk. The paper contains 50 theorems and several problems; some known theorems are included for the completion sake. Frequently the index k is dropped too; e.g. if \Phi is an order type such that \Phi \leq \lambda, |\Phi| > \aleph0 and if \alpha < \omega02, \beta < \omega02, \gamma < \omega1, then \Phi ––> (\omega0\gamma)2, \Phi ––> (\alpha,\beta)2 (Th. 5, and Zbl 048.28203, Theorems 5 and 7). The main problem is this: Is the relation \lambda ––> (\omega02,\omega02)2 true or false?
One of the main results reads (Th. 43): If r < s \leq b0, b1 ––> (s)rk then \alpha ––> (b0,b1)2 (this relation holds for order types as well as for cardinal numbers). If \phi is an order type > \aleph0 such that \omega1, \omega1^*\not \leq \phi and if \alpha < \omega 2, \beta < \omega2, \gamma < \omega1 then \phi ––> (\alpha,\alpha,\alpha)2\wedge (\alpha,\beta)2 \wedge (\omega,\gamma)2 \wedge (4,\alpha)3 (Th. 31). Let \alpha ––> (\beta, \gamma)2; let m be the initial ordinal of cardinality |\alpha|; then \beta < \omega0 \vee \gamma < \omega0 \vee \beta, \gamma \leq \alpha, m\vee \beta, \gamma \leq \alpha, m^* (Th. 19). If \alpha < \omega4 then \alpha (not)––> (3,\omega 2)2, \omega 4 ––> (3, \omega 2)2 (Th. 24). If r \geq 3, then \lambda (not)––> (\omega,\omega+2)r (Th. 27). |\lambda| (not)––> (\aleph1 \aleph1)r for r \geq 2 (Th. 30). For given r,k and \beta\nu (\nu < k), there exists an ordinal \alpha such that \alpha ––> (\beta0, \beta1,...,\beta\nu,...)rk (Cor. Th. 39). Moreover canonical partition relation as well as polarized partition relations are considered (\S\S 8,9).
Reviewer: G.Kurepa
Classif.: * 05D10 Ramsey theory
03E05 Combinatorial set theory (logic)
04A20 Combinatorial set theory
Index Words: set theory
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