Partitions and conservativity

TL;DR

This paper investigates the partition properties of certain ultrafilters, exploring their connections with conservativity and establishing new finite and infinite-exponent partition relations.

Contribution

It introduces new infinite-exponent partition relations for ultrafilters and links these with the concept of conservativity, expanding understanding of ultrafilter combinatorics.

Findings

Established finite-exponent partition relations for these ultrafilters.

Proved new infinite-exponent partition relations.

Connected partition properties with the notion of conservativity.

Abstract

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.