Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 257.04004
Autor: Erdös, Paul; Hajnal, András
Title: Ordinary partition relations for ordinal numbers. (In English)
Source: Period. Math. Hung. 1, 171-185 (1971).
Review: In this paper a number of positive and negative partition relations of the form \alpha ––> (\beta , \gamma)2 are established [see P. Erdös and R. Rado, Bull. Am. Math. Soc. 62, 427-489 (1956; Zbl 071.05105)]. E. Specker [Commentarii Math. Helvet. 31, 302-314 (1957; Zbl 080.03703)] proved that \omega 2 ––> (\omega 2,m)2 for m < \omega and A. Hajnal [Proc. Natl. Acad. Sci. USA 68, 142-144 (1971; Zbl 215.05201)], proved that the corresponding result for higher cardinals fails, by showing that, if \aleph\zeta is regular and GCH is assumed, then \omega2\zeta+1 (not)––> (\omega2\zeta+1,3)2.
This negative result is extended here and some complementary positive theorems are proved. Thus, if \aleph\zeta is regular, GCH is assumed, k,t < \omega, m = (t+1)(k+1) and \mu < \beta = \omegak+2\zeta+1, then \omega m\zeta+1 (not)––> (\beta,t+2)2, \omegam\zeta+1 ––> (\mu,t+2)2, \omega m+1\zeta+1 (not)––> (\beta+1,t+2)2 and \omegam+1\zeta+1 ––> (\beta,t+2)2. This leaves some questions open.
For example, the authors ask if \omega21 ––> (\omega1\tau,4)2 for all \tau < \omega1. C. C. Chang [J. Comb. Theory, Ser. A 12, 396-452 (1972; Zbl 266.04003)], proved that \omega\omega ––> (\omega\omega ,3)2 (and it is known that \omega\omega ––> (\omega\omega ,m)2 for all m < \omega). In contrast to this, it is shown here (assuming GCH) that \sigma (not)––> (\omega\omega1,3)2 for all \sigma < \omega2. It is not known if the analogous result \sigma (not)––> (\omega\omega12,3)2 holds for all \sigma < \omega3.
Reviewer: E.C.Milner
Classif.: * 04A20 Combinatorial set theory
05A17 Partitions of integres (combinatorics)
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