LAUSR

A hierarchy of topological Ramsey spaces $${\mathcal{R}_\alpha}$$Rα ($${\alpha < \omega_1}$$α<ω1), generalizing the Ellentuck space, were built by Dobrinen and Todorcevic in order to completely classify certain equivalent classes of ultrafilters Tukey (resp. Rudin–Keisler) below $${\mathcal{U}_\alpha}$$Uα$${(\alpha < \omega_1)}$$(α<ω1), where $${\mathcal{U}_\alpha}$$Uα are ultrafilters constructed by Laflamme satisfying certain partition properties and have complete combinatorics over the Solovay model. We show that Nash–Williams, or Ramsey ultrafilters in these spaces are preserved under countable-support side-by-side Sacks forcing. This is achieved by proving a parametrized theorem for these spaces, and showing that Nash–Williams ultrafilters localizes the theorem. We also show that every Nash–Williams ultrafilter in $${\mathcal{R}_\alpha}$$Rα is selective.