# Natasha Dobrinen: Ramsey classification theorems for a new class of topological Ramsey spaces, and their applications in the Tukey theory of ultrafilters.

Friday, August 3 at 1:30pm
Fields institute, Room 210

Speaker: Natasha Dobrinen (Denver)

Title: Ramsey classification theorems for a new class of topological Ramsey spaces, and their applications in the Tukey theory of ultrafilters.

Abstract: The Erdos-Rado Theorem and the Pudlak-Rodl Theorem are generalizations of the well-known Ramsey Theorem. The generalizations are from finitely many colors to infinitely many colors (Erdos-Rado), and furthermore to barriers on $\mathbb{N}$ in place of $[\mathbb{N}]^k$ (Pudlak-Rodl).
These theorems are canonization theorems on the Ellentuck space.

In this talk, we present some joint work with Todorcevic. We construct a new class of topological Ramsey spaces $\mathcal{R}_{\alpha}$, $\alpha<\omega_1$. These spaces were inspired by some work of Laflamme, regarding complete combinatorics for ultrafilters satisfying particular partition properties. The space $\mathcal{R}_1$ is minimal in complexity over the Ellentuck space, and each $\mathcal{R}_{\alpha}$ is minimal in complexity over the collection of $\mathcal{R}_{\beta}$, $\beta<\alpha$. To each space is associated an ultrafilter $\mathcal{U}_{\alpha}$ which is “Ramsey for $\mathcal{R}_{\alpha}$”. In particular, $\mathcal{U}_1$ is weakly Ramsey but not Ramsey.

We prove the analogues of the Erdos-Rado and Pudlak-Rodl Theorems for all $\mathcal{R}_{\alpha}$. These are then used to completely classify all the Rudin-Keisler (isomorphism) classes of ultrafilters which are Tukey reducible to the ultrafilters $\mathcal{U}_{\alpha}$. These are exactly the countable iterations of Fubini products of ultrafilters from among an easily definable countable collection of ultrafilters.