The relation of rapid ultrafilters and Q-points to van der Waerden ideal

TL;DR

This paper explores the distinctions between rapid ultrafilters and Q-points in relation to the van der Waerden ideal, highlighting differences in their intersections and constructing specific ultrafilters under certain axioms.

Contribution

It identifies a key difference in intersection properties with the van der Waerden ideal and constructs a W-ultrafilter that is not a Q-point assuming Martin's axiom.

Findings

Rapid ultrafilters can have empty intersection with the van der Waerden ideal.

Every Q-point has a non-empty intersection with the van der Waerden ideal.

Under Martin's axiom, a W-ultrafilter that is not a Q-point can be constructed.

Abstract

We point out one of the differences between rapid ultrafilters and Q-points: Rapid ultrafilters may have empty intersection with van der Waerden ideal, whereas every Q-point has a non-empty intersection with van der Waerden ideal. Assuming Martin's axiom for countable posets we also construct a W-ultrafilter which is not a Q-point.