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2017 Selective and Ramsey Ultrafilters on G-spaces
Oleksandr Petrenko, Igor Protasov
Notre Dame J. Formal Logic 58(3): 453-459 (2017). DOI: 10.1215/00294527-3839090


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.


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Oleksandr Petrenko. Igor Protasov. "Selective and Ramsey Ultrafilters on G-spaces." Notre Dame J. Formal Logic 58 (3) 453 - 459, 2017.


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

zbMATH: 1368.05150
MathSciNet: MR3681104
Digital Object Identifier: 10.1215/00294527-3839090

Primary: 05D10 , X001
Secondary: 54H15

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

Rights: Copyright © 2017 University of Notre Dame


Vol.58 • No. 3 • 2017
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