The polarized Ramsey’s theorem Article

First Online: 31 October 2008 Received: 06 May 2008 Revised: 19 July 2008 DOI :
10.1007/s00153-008-0108-0

Cite this article as: Dzhafarov, D.D. & Hirst, J.L. Arch. Math. Logic (2009) 48: 141. doi:10.1007/s00153-008-0108-0
Abstract We study the effective and proof-theoretic content of the polarized Ramsey’s theorem, a variant of Ramsey’s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields a new characterization of Ramsey’s theorem in all exponents, and produces several combinatorial principles which, modulo bounding for \({\Sigma^0_2}\) formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey’s theorem for pairs.

Mathematics Subject Classification (2000) 03B30 03F35 03D80 05D10 We are grateful to D. Hirschfeldt, A. Montalbán, and R. Soare for making our collaboration possible and for helpful comments and suggestions. We thank J. Schmerl for first bringing the subject of polarized partitions to our attention and J. Mileti for his generous insights. We also thank one anonymous referee for valuable observations and corrections. The first author was partially supported by an NSF Graduate Research Fellowship.

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Authors and Affiliations 1. University of Chicago Chicago USA 2. Appalachian State University Boone USA