Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 363.04009
Autor: Cates, M.; Erdös, Paul; Hindman, N.; Rothschild, B.
Title: Partition theorems for subsets of vector spaces. (In English)
Source: J. Comb. Theory, Ser. A 20, 279-291 (1976).
Review: Partition theorems are considered also in vector spaces (cf. the papers of P.Erdös and R.Rado in Bull. Am. Math. Soc. 62, 427-489 (1956; Zbl 071.05101) and of P.Erdös, A. Hajnal and R.Rado in Acta Math. Acad. Sci. Hung. 16, 93-196 (1965; Zbl 158.26603)).
In particular, for any given quadruple (\alpha,\beta,\gamma,\delta) of cardinals let <\alpha> ––> <\beta>\gamma\delta mean the following: Whenever V is an \alpha-dimensional vector space over GF(2) and V = \bigcup\sigma < \gammaA\sigma there are some U in [V]\beta and some \sigma < \gamma such that if 1 \leq t < \delta and W in [U] then \Sigma W = A\sigma. In section 2 the authors prove 19 lemmas; e.g. L.r: 3(a): If \beta, \gamma, \delta < \omega, then there exists the least integer N(\beta, \gamma, \delta) such that < N(\beta,\gamma,\delta)> ––> <\omega>\gamma\omega. L.r: 9: If \beta is a regular cardinal > \omega and <\beta> ––> <\beta>\\gamma3 then \beta ––> (\beta)\gamma2; L.2: 10: GCH implies that every infinite nonlimit cardinal \beta satisfies <\beta> (not)––> <\beta>23.
The main result (Th. 3: 1): Assume the GCH and that there is no inaccessible cardinal > \omega. Exclude the possibility that any of the conditions: (a), (b) or (c) holds: at3 (a) \delta < \omega, \beta < \omega, \gamma = \aleph\rho and \aleph\rho+\delta-1 \leq \alpha < \aleph\rho+2\beta- 1; \&(b) \delta = 4, \gamma < \omega, \beta = \aleph\rho > cf(\beta) = \omega and \alpha < \aleph\rho+t(\gamma), t(\gamma): = 2sumi = 0\gamma\frac{r!}{i!}-1;
(c) \delta = 4, \gamma < \omega, cf(\beta) > \omega, \beta = \aleph\rho and \aleph{\rho} < \alpha < \aleph\rho+t(\gamma). Then <\alpha> ––> <\beta>\gamma\delta holds if and only if one of the following 10 statements holds:
(1) \gamma = 1; (2) \delta = 2 and \beta = 1;
(3) \delta = 2, \alpha \geq \omega, \gamma < \alpha, and \beta < \alpha;
(4) \delta = 2, \alpha \geq \omega, \gamma < cf(\alpha), and \beta = \alpha;
(5) \delta = 3, \beta < \alpha, \alpha > \omega , and \gamma^+< \alpha;
(6) \delta = 3, \beta = \alpha > \omega, cf(\alpha) = \omega, and \gamma < \omega;
(7) \delta = 4, \gamma < \omega, \beta = \aleph\rho, and \alpha \geq \aleph\rho+t(\gamma);
(8) \beta < \omega, \gamma < \omega, \alpha \geq N(\beta,\gamma,\delta);
(9) \beta < \omega, \gamma = \aleph\rho, and \alpha \geq \aleph{\rho+2\beta-1};
(10) \beta = \omega and \gamma < \omega.
Reviewer: D.Kurepa
Classif.: * 04A20 Combinatorial set theory
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