PreprintCC BY 4.0arXiv
Combinatorics of 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 Schur ultrafilter on ℤ that is not infinitary Schur. Moreover, assuming the Continuum Hypothesis, we establish the existence of Schur P-points in β(ℤ). On the other hand, it is consistent with ZFC that there exist P-points in β(ℤ), but none of them are Schur. Also, we extend the result of Fernández-Bretón, Navarro-Castillo, and Soria-Rojas by showing that no Schur ultrafilter on ℤ is a Q-point.
Related
BibTeX
@misc{bardyla2026combinatorics,
title = {{Combinatorics of Schur ultrafilters}},
author = {S. Bardyla},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.17411},
eprint = {2605.17411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17411},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Combinatorics of Schur ultrafilters
AU - S. Bardyla
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.17411
UR - https://arxiv.org/abs/2605.17411
AB - In this paper, we provide a combinatorial characterization of the elements of Schur ultrafilters on countable commutative groups. Using this characterization, we construct a Schur ultrafilter on ℤ that is not infinitary Schur. Moreover, assuming the Continuum Hypothesis, we establish the existence of Schur P-points in β(ℤ). On the other hand, it is consistent with ZFC that there exist P-points in β(ℤ), but none of them are Schur. Also, we extend the result of Fernández-Bretón, Navarro-Castillo, and Soria-Rojas by showing that no Schur ultrafilter on ℤ is a Q-point.
ER -