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September 2013 Comparisons of polychromatic and monochromatic Ramsey theory
Justin Palumbo
J. Symbolic Logic 78(3): 951-968 (September 2013). DOI: 10.2178/jsl.7803130

Abstract

We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erdős and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various special classes of ultrafilters, showing for example that every rainbow Ramsey ultrafilter is nowhere dense but rainbow Ramsey ultrafilters need not be rapid. This entails comparison of the polychromatic and monochromatic Ramsey theorems as combinatorial principles on $\omega$.

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Justin Palumbo. "Comparisons of polychromatic and monochromatic Ramsey theory." J. Symbolic Logic 78 (3) 951 - 968, September 2013. https://doi.org/10.2178/jsl.7803130

Information

Published: September 2013
First available in Project Euclid: 6 January 2014

zbMATH: 1345.03088
MathSciNet: MR3135506
Digital Object Identifier: 10.2178/jsl.7803130

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 3 • September 2013
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