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 U on X is called G-selective if, for any G-invariant partition P of X, either one cell of P is a member of U, or there is a member of U which meets each cell of 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 U on ω is selective if and only if U is G-selective with respect to the action of any countable group G of permutations of ω.

A free ultrafilter U on X is called G-Ramsey if, for any G-invariant coloring χ:[X]2{0,1}, there is UU such that [U]2 is χ-monochromatic. We show that each G-Ramsey ultrafilter on X is G-selective. Additional theorems give a lot of examples of ultrafilters on Z that are Z-selective but not 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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH 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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