An infinitary Ramsey property

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Abstract

Mitchell, W.J., An infinitary Ramsey property, Annals of Pure and Applied Logic 57 (1992) 151–160. We prove that the consistency of a measurable cardinal implies the consistency of a cardinal κ>(2ω)+ satisfying the partition relations κ

(ω)ω
and κ
(ω)ωregressive. This result follows work of Spector which uses the same hypothesis to prove the consistency of ω1
(ω)ω. We also give some examples of partition relations which can be proved for ω1 using the methods of Spector but cannot be proved for cardinals κ>(2ω)+ without a much stronger hypothesis.

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Much of the research was done while the author was visiting UCLA, The California Institute of Technology and The Hebrew University. While at the Hebrew University he was partly supported by the Lady Davis Foundation. The work was also partially supported by grant number DMS-8614447 from the National Science Foundation.