Combinatorics of Schur ultrafilters

TL;DR

This paper characterizes Schur ultrafilters on countable groups, constructs a specific ultrafilter on integers, and under CH, proves the existence of a Schur P-point, advancing ultrafilter theory.

Contribution

It provides a combinatorial characterization of Schur ultrafilters and constructs new examples, including a free Schur ultrafilter not infinitary Schur and a Schur P-point under CH.

Findings

Constructed a free Schur ultrafilter on d7 with specific properties.

Established the existence of a Schur P-point on d7 under the Continuum Hypothesis.

Provided a combinatorial framework for understanding Schur ultrafilters.

Abstract

In this paper, we provide a combinatorial characterization of the elements of Schur ultrafilters on countable commutative groups. Using this characterization, we construct a free Schur ultrafilter on $\mathbb Z$ that is not infinitary Schur. Moreover, assuming the Continuum Hypothesis, we establish the existence of a free Schur P-point on $\mathbb Z$.