## Journal of Symbolic Logic

### Comparisons of polychromatic and monochromatic Ramsey theory

Justin Palumbo

#### 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 $\omega$.

#### Article information

Source
J. Symbolic Logic, Volume 78, Issue 3 (2013), 951-968.

Dates
First available in Project Euclid: 6 January 2014

https://projecteuclid.org/euclid.jsl/1389032283

Digital Object Identifier
doi:10.2178/jsl.7803130

Mathematical Reviews number (MathSciNet)
MR3135506

Zentralblatt MATH identifier
1345.03088

#### Citation

Palumbo, Justin. Comparisons of polychromatic and monochromatic Ramsey theory. J. Symbolic Logic 78 (2013), no. 3, 951--968. doi:10.2178/jsl.7803130. https://projecteuclid.org/euclid.jsl/1389032283