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.
"Selective and Ramsey Ultrafilters on -spaces." Notre Dame J. Formal Logic 58 (3) 453 - 459, 2017. https://doi.org/10.1215/00294527-3839090