Natasha Dobrinen

A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This extends the Pudlak-Rodl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our canonization theorem to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U1: Every ultrafilter which is Tukey reducible to U1 is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of ultrafilters. Moreover, we show that there is exactly one Tukey type of nonprincipal ultrafilters strictly below that of U1, namely the Tukey type of a Ramsey ultrafilter.

Homogeneous iteration and measure one covering relative to HOD

Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. In fact it is consistent that there is a superstrong cardinal and for every regular cardinal κ, κ+ is greater than κ+ of HOD. The proof uses a very general lemma showing that homogeneity is preserved through certain reverse Easton iterations.

Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not

Rainbow Ramsey simple structures

k+1

Errata to “Games and general distributive laws in Boolean algebras”

k+1-arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra

The hyper-weak distributive law and a related game in Boolean algebras

\mathcal {P}(n)