Comparisons of polychromatic and monochromatic Ramsey theory

TL;DR

This paper compares polychromatic and monochromatic Ramsey theories, exploring their logical relationships, introducing rainbow Ramsey ultrafilters, and connecting these concepts to classical set-theoretic cardinal characteristics.

Contribution

It establishes new independence results, introduces rainbow Ramsey ultrafilters, and links polychromatic Ramsey theory to classical set-theoretic invariants.

Findings

Rainbow Ramsey theorem does not follow from ZF.

Axiom of choice affects infinite partition relations.

Rainbow Ramsey ultrafilters are nowhere dense but not necessarily rapid.

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 Erd{\"o}s 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 polychromatic and monochromatic Ramsey theory in some countable combinatorial settings. Finally we give new characterizations of the bounding and dominating numbers and the covering and uniformity numbers of the meager ideal which are in the spirit of polychromatic Ramsey theory.