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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
Show Scanned Page ##### MSC:
 05D10 Ramsey theory 03E05 Other combinatorial set theory
set theory
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