## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic (2017), 7 pages.

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

Oleksandr Petrenko and Igor Protasov

#### 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 $\omega$.
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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.
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*Article information*

*Article information*

**Source**

Notre Dame J. Formal Logic (2017), 7 pages.

**Dates**

Received: 27 June 2012

Accepted: 27 August 2014

First available in Project Euclid: 19 April 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00294527-3839090

**Subjects**

Primary: X001 05D10: Ramsey theory [See also 05C55]

Secondary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]

**Keywords**

$G$-space $G$-selective and $G$-Ramsey ultrafilters Stone–Čech compactification

*Citation*

*Citation*

*Petrenko, Oleksandr; Protasov, Igor. Selective and Ramsey Ultrafilters on
G
-spaces.
Notre Dame J. Formal Logic, advance publication, 19 April 2017. doi: 10.1215/00294527-3839090. http://projecteuclid.org/euclid.ndjfl/1492567511*