Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 58, Number 3 (2017), 453-459.
Selective and Ramsey Ultrafilters on -spaces
Let be a group, and let be an infinite transitive -space. A free ultrafilter on is called -selective if, for any -invariant partition of , either one cell of is a member of , or there is a member of which meets each cell of in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a -selective ultrafilter on . We describe all -spaces such that each free ultrafilter on is -selective, and we prove that a free ultrafilter on is selective if and only if is -selective with respect to the action of any countable group of permutations of .
A free ultrafilter on is called -Ramsey if, for any -invariant coloring , there is such that is -monochromatic. We show that each -Ramsey ultrafilter on is -selective. Additional theorems give a lot of examples of ultrafilters on that are -selective but not -Ramsey.
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
Permanent link to this document
Digital Object Identifier
Primary: X001 05D10: Ramsey theory [See also 05C55]
Secondary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]
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