  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.
