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.
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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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Dzhafarov, D.D., Hirst, J.L. The polarized Ramsey’s theorem. Arch. Math. Logic 48, 141–157 (2009). https://doi.org/10.1007/s00153-008-0108-0
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DOI: https://doi.org/10.1007/s00153-008-0108-0