  We study the partition properties enjoyed by the "next best thing to a
P-point'' ultrafilters introduced recently in joint work with Dobrinen and
Raghavan. That work established some finite-exponent partition relations, and
we now analyze the connections between these relations for different exponents
and the notion of conservativity introduced much earlier by Phillips. In
addition, we establish some infinite-exponent partition relations for these
ultrafilters and also for sums of non-isomorphic selective ultrafilters indexed
by selective ultrafilters.
