$\mathbb Z$-Ramsey ultrafilters

Abstract

We study $\mathbb{Z}$-Ramsey ultrafilters, ultrafilters containing witnesses to every shift-invariant instance of Ramsey's theorem. We prove that it is consistent that there are no $\mathbb Z$-Ramsey ultrafilters. We also prove that every $(\mathbb{Z},3)$-Ramsey ultrafilter, as well as every $\mathbb{Z}$-Ramsey P-point, is selective. Further, we exhibit a generic extension -- using quotient algebras of the form \(\mathcal{P}(\mathbb{Z})/\mathcal{I}\) for certain \(F_{σ}\)-ideals -- that contains P-points that are not \(\mathbb{Z}\)-Ramsey ultrafilters, thereby addressing open questions raised by Petrenko and Protasov.