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Comparisons of Polychromatic and Monochromatic Ramsey Theory

Published online by Cambridge University Press:  12 March 2014

Justin Palumbo
Justin Palumbo*
Affiliation:
Department of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095, USA, E-mail: justinpa@math.ucla.edu
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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 ω.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 951 - 968
Copyright
Copyright © Association for Symbolic Logic 2013

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