Notre Dame Journal of Formal Logic

Selective and Ramsey Ultrafilters on $G$-spaces

Oleksandr Petrenko and Igor Protasov

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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

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

Received: 27 June 2012
Accepted: 27 August 2014
First available in Project Euclid: 19 April 2017

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Digital Object Identifier

Primary: X001 05D10: Ramsey theory [See also 05C55]
Secondary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]

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


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.

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