Ramsey theory and topological dynamics of 0-dimensional flows
Abstract
We introduce several Ramsey-theoretic properties of 0-dimensional ambits and obtain their dynamical characterizations. In consequence, we obtain Ramsey-theoretic criteria for triviality and for profiniteness of some important invariants (in particular, of the Ellis groups) of the ambits in question. This yields a criterion for the structural property that each distal minimal factor of the given ambit is a profinite flow. Another result is a criterion for metrizability of the minimal left ideals in the Ellis semigroup. Then we study three specializations of the above abstract context: to first order theories, to definable groups, and to the classical Kechris-Pestov-Todorčević theory, recovering known and obtaining new results in each of these contexts.
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BibTeX
@misc{krupinski2026ramsey,
title = {{Ramsey theory and topological dynamics of 0-dimensional flows}},
author = {Krzysztof Krupiński and Junguk Lee and Slavko Moconja},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25132},
eprint = {2609.25132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25132},
}
RIS
TY - UNPB
TI - Ramsey theory and topological dynamics of 0-dimensional flows
AU - Krzysztof Krupiński
AU - Junguk Lee
AU - Slavko Moconja
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.25132
UR - https://arxiv.org/abs/2609.25132
AB - We introduce several Ramsey-theoretic properties of 0-dimensional ambits and obtain their dynamical characterizations. In consequence, we obtain Ramsey-theoretic criteria for triviality and for profiniteness of some important invariants (in particular, of the Ellis groups) of the ambits in question. This yields a criterion for the structural property that each distal minimal factor of the given ambit is a profinite flow. Another result is a criterion for metrizability of the minimal left ideals in the Ellis semigroup. Then we study three specializations of the above abstract context: to first order theories, to definable groups, and to the classical Kechris-Pestov-Todorčević theory, recovering known and obtaining new results in each of these contexts.
ER -