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A remark on partition relations for infinite ordinals with an application to finite combinatorics. (English) Zbl 0635.03042

Logic and combinatorics, Proc. AMS-IMS-SIAM Conf., Arcata/Calif. 1985, Contemp. Math. 65, 157-167 (1987).
[For the entire collection see Zbl 0619.00005.]
In this paper the authors prove a number of partition relations for ordinal numbers, which settle several long standing open problems. Their main results are the following (where multiplication and exponentiation mean ordinal multiplication and ordinal exponentiation).
(1) If \(\kappa\) is a regular cardinal and \(2^{\kappa}=\kappa\) \(+\) then (\(\kappa\) \(+)\) \(2\nrightarrow (\kappa\) \(+\cdot \kappa,4)\) 2.
(2) If \(\kappa\) is a singular cardinal with \(\tau =cf(\kappa)\) and \(2^{\kappa}=\kappa\) \(+\) then \(\kappa\) \(+\cdot \tau \nrightarrow (\kappa\) \(+\cdot \tau,3)\) 2.
(3) If \(\kappa\) is a regular cardinal and \(\kappa^{<\kappa}=\kappa\) then (\(\kappa\) \(+)\) \(2\to (\kappa\) \(+\cdot \kappa,3,3)\) 2.
Reviewer: N.H.Williams

MSC:

03E05 Other combinatorial set theory
03E10 Ordinal and cardinal numbers

Citations:

Zbl 0619.00005