On sequentially Ramsey sets (Q2839284)

If \(\mathcal F\) is a free ultrafilter on the set \(\omega\) of integers (natural numbers), then the set \(^\omega\omega\) of all infinite sequences of integers can be equipped with the topology generated by bodies of Laver trees such that immediate successors of any node, except for those belonging to the stem, consist of a set belonging to \(\mathcal F\). Sets with the Baire property with respect to this topology are called sequentially completely Ramsey. The family of all these sets is denoted by \(S_{\mathcal F}CR\), while \(S_{\mathcal F}CR^0\) denotes the family of all hereditary sequentially completely Ramsey sets (nowhere dense). The additivity, cofinality, covering and uniformity of the \(\sigma\)-ideal \(S_{\mathcal F}CR^0\) are compared with the minimal size of an unbounded subfamily of \(^\omega\omega\) and with the minimal size of a dominating subfamily of \(^\omega\omega\). More general versions of similarly introduced topologies were examined in the papers [\textit{A. Błaszczyk}, Topol. Proc. 41, 65--84 (2013; Zbl 1286.54031)] or also [\textit{A. Błaszczyk} and \textit{A. Brzeska}, Colloq. Math. 131, No. 1, 89--98 (2013; Zbl 1280.54021)].NEWLINENEWLINENote that, Question 3.6 [{If \(add (S_{\mathcal F}CR^0)= \lambda> \omega\), is then \(\mathcal F\)} a \(P(\lambda)\)-ultrafilter?] has a negative consistent answer. Indeed, there are always ultrafilters which are not \(p\)-points. If the continuum hypothesis holds and an ultrafilter \(\mathcal F\) is not \(p\)-point, then \(add (S_{\mathcal F}CR^0)= \omega_1 =\mathfrak c \).