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A partition calculus in set theory. (English) Zbl 0071.05105
The memoir is a natural sequel of some previous articles (Zbl 0038.15301; Zbl 0048.28203; Zbl 0051.04003; Zbl 0055.04903) initiated by F. P. Ramsey in 1930 [Proc. Lond. Math. Soc. (2) 30, 264–286 (1930; JFM 55.0032.04)], see also D. Kurepa [C. R. Soc. Sci. Varsovie, Cl. III 32, 61–67 (1939), reprinted in Periodicum Math.-Phys. Astron., II. Ser. 14, 205–210 (1959; Zbl 0094.03301); 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 \(a\to [b_0,b_1,...,b_\nu,...]^r_k\) is said to hold provided for a set \(S\) of cardinality \(a\) and for every partition of the set \([S]^r\): \(S^r= \bigcup_{r<k} K_\nu\) 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 \(b_0\) the corresponding relation is denoted \(a \to (b_0)^r_k\). 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 \(\Phi \leq \lambda,\) \(|\Phi| > \aleph_0\) and if \(\alpha < \omega_02\), \(\beta<\omega_0^2\), \(\gamma < \omega_1\), then \(\Phi \to (\omega_0\gamma)^2\), \(\Phi \to (\alpha,\beta)^2\) (Th. 5, and Zbl 0048.28203, Theorems 5 and 7). The main problem is this: Is the relation \(\lambda \to (\omega_02,\omega_0^2)^2\) true or false?
One of the main results reads (Th. 43): If \(r < s\leq b_0\), \(b_1 \to (s)^r_k\) then \(\alpha \to (b_0,b_1)^2\) (this relation holds for order types as well as for cardinal numbers). If \(\varphi\) is an order type \(>\aleph_0\) such that \(\omega_1, \omega_1^*\not\leq \varphi\) and if \(\alpha < \omega 2\), \(\beta < \omega^2\), \(\gamma < \omega_1\) then \(\varphi \to (\alpha,\alpha,\alpha)^2\wedge (\alpha,\beta)^2 \wedge (\omega,\gamma)^2 \wedge (4,\alpha)^3\) (Th. 31). Let \(\alpha \to (\beta, \gamma)^2\); let \(m\) be the initial ordinal of cardinality \(|\alpha|\); then \(\beta < \omega_0 \vee \gamma< \omega_0 \vee \beta, \gamma \leq \alpha\), \(m\vee \beta, \gamma \leq \alpha, m^*\) (Th. 19). If \(\alpha < \omega_4\) then \(\alpha \nrightarrow (3,\omega 2)^2\), \(\omega 4 \to (3, \omega 2)^2\) (Th. 24). If \(r\geq 3\), then \(\lambda \nrightarrow (\omega,\omega+2)^r\) (Th. 27). \(|\lambda| \nrightarrow (\aleph_1 \aleph_1)^r\) for \(r \geq 2\) (Th. 30). For given \(r,k\) and \(\beta_\nu\) (\(\nu < k\)), there exists an ordinal \(\alpha\) such that \(\alpha \to (\beta_0, \beta_1,...,\beta_\nu,...)^r_k\) (Cor. Th. 39). Moreover canonical partition relation as well as polarized partition relations are considered \((\S\S 8,9)\).
Reviewer: G.Kurepa

05D10 Ramsey theory
03E05 Other combinatorial set theory
set theory
Full Text: DOI
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