## Notre Dame Journal of Formal Logic

### Selective and Ramsey Ultrafilters on $G$-spaces

#### Abstract

Let $G$ be a group, and let $X$ be an infinite transitive $G$-space. A free ultrafilter $\mathcal{U}$ on $X$ is called $G$-selective if, for any $G$-invariant partition $\mathcal{P}$ of $X$, either one cell of $\mathcal{P}$ is a member of $\mathcal{U}$, or there is a member of $\mathcal{U}$ which meets each cell of $\mathcal{P}$ in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a $G$-selective ultrafilter on $X$. We describe all $G$-spaces $X$ such that each free ultrafilter on $X$ is $G$-selective, and we prove that a free ultrafilter $\mathcal{U}$ on $\omega$ is selective if and only if $\mathcal{U}$ is $G$-selective with respect to the action of any countable group $G$ of permutations of $\omega$.

A free ultrafilter $\mathcal{U}$ on $X$ is called $G$-Ramsey if, for any $G$-invariant coloring $\chi:[X]^{2}\to\{0,1\}$, there is $U\in\mathcal{U}$ such that $[U]^{2}$ is $\chi$-monochromatic. We show that each $G$-Ramsey ultrafilter on $X$ is $G$-selective. Additional theorems give a lot of examples of ultrafilters on $\mathbb{Z}$ that are $\mathbb{Z}$-selective but not $\mathbb{Z}$-Ramsey.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 3 (2017), 453-459.

Dates
Accepted: 27 August 2014
First available in Project Euclid: 19 April 2017

https://projecteuclid.org/euclid.ndjfl/1492567511

Digital Object Identifier
doi:10.1215/00294527-3839090

Mathematical Reviews number (MathSciNet)
MR3681104

Zentralblatt MATH identifier
1368.05150

#### Citation

Petrenko, Oleksandr; Protasov, Igor. Selective and Ramsey Ultrafilters on $G$ -spaces. Notre Dame J. Formal Logic 58 (2017), no. 3, 453--459. doi:10.1215/00294527-3839090. https://projecteuclid.org/euclid.ndjfl/1492567511

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