Local Ramsey theory. An abstract approach

Topological Ramsey Spaces Dense in Forcings

“…The second and stronger property satisfied by ultrafilters forced by (R, ≤ * ) is the analogue of selectivity, with respect to the topological Ramsey space. In [9], the general definition of semiselective coideal is presented. Here we shall only concentrate on when the generic filter U is selective.…”

“…Theorem 23 (Di Prisco/Mijares/Nieto, [9]). If there exists a supercompact cardinal and G ⊆ R is generic for (R, ≤ * ), then all definable subsets of R are G-Ramsey.…”

“…Theorem 24 (Di Prisco/Mijares/Nieto, [9]). Let (R, ≤, r) be a topological Ramsey space and U ⊆ R be a selective ultrafilter in a transitive model M of ZF + DCR.…”

Creature forcing and topological Ramsey spaces

“…In particular, it makes investigations of forcing over L(R) reasonable, as all subsets of the space in L(R) are Ramsey. Further, by work of Di Prisco, Mijares, and Nieto in [4], in the presence of a supercompact cardinal, the generic ultrafilter forced by a topological Ramsey space, partially ordered by almost reduction, has complete combinatorics in over L(R). Having at one's disposal the Abstract Ellentuck Theorem or the Abstract Nash-Williams Theorem aids in proving canonical equivalence relations on fronts and barriers, in the vein of Pudlák and Rödl [12].…”