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Ordinary partition relations for ordinal numbers. (English) Zbl 0257.04004

In this paper a number of positive and negative partition relations of the form \(\alpha \to (\beta , \gamma)^2\) are established [see P. Erdős and R. Rado, Bull. Am. Math. Soc. 62, 427-489 (1956; Zbl 0071.05105)]. E. Specker [Commentarii Math. Helvet. 31, 302-314 (1957; Zbl 0080.03703)] proved that \(\omega ^2 \to (\omega ^2,m)^2\) for \(m< \omega\) and A. Hajnal [Proc. Natl. Acad. Sci. USA 68, 142-144 (1971; Zbl 0215.05201)], proved that the corresponding result for higher cardinals fails, by showing that, if \(\aleph_\zeta\) is regular and GCH is assumed, then \(\omega^2_{\zeta +1} \nrightarrow (\omega^2_{\zeta +1},3)^2\).
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 \(\mu < \beta = \omega^{k+2}_{\zeta +1}\), then \(\omega ^m_{\zeta +1} \nrightarrow (\beta,t+2)^2\), \(\omega^m_{\zeta+1} \to (\mu,t+2)^2\), \(\omega ^{m+1}_{\zeta+1} \nrightarrow (\beta+1,t+2)^2\) and \(\omega^{m+1}_{\zeta+1} \to (\beta,t+2)^2\). This leaves some questions open.
For example, the authors ask if \(\omega^2_1 \to (\omega_1\tau,4)^2\) for all \(\tau < \omega_1\). C. C. Chang [J. Comb. Theory, Ser. A 12, 396-452 (1972; Zbl 0266.04003)], proved that \(\omega^\omega \to (\omega^\omega ,3)^2\) (and it is known that \(\omega^\omega \to (\omega^\omega ,m)^2\) for all \(m< \omega\)). In contrast to this, it is shown here (assuming GCH) that \(\sigma \nrightarrow (\omega^\omega_1,3)^2\) for all \(\sigma < \omega_2\). It is not known if the analogous result \(\sigma \nrightarrow (\omega^{\omega_1}_2,3)^2\) holds for all \(\sigma < \omega_3\).
Reviewer: E.C.Milner

MSC:

03E05 Other combinatorial set theory
05A17 Combinatorial aspects of partitions of integers
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[1] Erdős, P.; Rado, R., Combinatorial theorems on classification of subsets of a given set, Proc. London Math. Soc., 2, 3, 417-439 (1952) · Zbl 0048.28203
[2] Erdős, P.; Rado, R., A partition calculus in set theory, Bull. Amer. Math. Soc., 62, 427-489 (1956) · Zbl 0071.05105
[3] Erdős, P.; Hajnal, A.; Rado, R., Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hung., 16, 93-196 (1965) · Zbl 0158.26603
[4] P. Erdős andA. Hajnal, Unsolved problems in set theory,Axiomatic Set Theory, Proc. of Symp. in pure math.13 (1971) Part 1. · Zbl 0228.04001
[5] Hajnal, A., A negative partition relation, Proc. Nat. Acad., 68, 142-144 (1971) · Zbl 0215.05201
[6] Erdős, P.; Rado, R., Partition relations and transitivity domains of binary relations, J. London Math. Soc., 42, 624-633 (1967) · Zbl 0204.00905
[7] Milner, E. C.; Rado, R., The pigeon-hole principle for ordinal numbers, proc. London Math. Soc., 15, 3, 750-768 (1965) · Zbl 0145.24501
[8] P. Erdős, A. Hajnal andE. C. Milner, Set mappings and polarized partitions,Combinatorial Theory and its Appl. (Colloqu. Math. J. Bolyai4), Amsterdam-London, 1970, 327-363. · Zbl 0215.32903
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