#### Date of Award

1-1-2014

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Natasha Dobrinen

#### Keywords

Dedekind Cuts, Selective, Topological Ramsey Spaces, Tukey Theory, Ultrafilters, Ultrapowers

#### Abstract

This dissertation makes contributions to the areas of combinatorial set theory, the model theory of arithmetic, and the Tukey theory of ultrafilters. The main results are broken into three parts.

In the first part, we identify some new partition relations among finite trees and use them to answer an open question of Dobrinen; namely, "for n < omega, are the notions of Ramsey for Rn and selective for Rn equivalent?" We show that for each n < omega, it is consistent with ZFC that there exists a selective for Rn ultrafilter which is not Ramsey for Rn.

In the second part, we extend results of Blass concerning Dedekind cuts associated to ultrafilter mappings from p-point and weakly-Ramsey ultrafilters to ultrafilter mappings from Ramsey for R1 ultrafilters. Blass associates to each ultrafilter U on a countable set X and each function g with domain X a Dedekind cut in the model of arithmetic given by the ultrapower omega^ran(g)/g(U). We characterize, under the continuum hypothesis, the cuts obtainable from an ultrafilter mapping from a Ramsey for R1 ultrafilter. We also show that the only cut obtainable for ultrafilter mappings between p-points, which are Tukey reducible to a given Ramsey for R1 ultrafilter, is the standard cut consisting of equivalence classes of constant sequences. These results imply new existence theorems for various special kinds of ultrafilters.

In final part of the dissertation, we extend results of Dobrinen and Todorcevic concerning the canonical Ramsey theory of R1 to the space H2 given by forming the product of the space R1 with itself. These results imply new existence theorems for initial Tukey structures of nonprincipal ultrafilters. These results shed light on the following open question of Dobrinen concerning the Tukey theory of ultrafilters, "what are the possible initial Tukey structures for ultrafilters on a countable base set?" In particular, we show for the first time that it is consistent with ZFC that the four-element Boolean algebra appears as an initial Tukey structure.

#### Recommended Citation

Trujillo, Timothy Onofre, "Topological Ramsey spaces, associated ultrafilters, and their applications to the Tukey theory of ultrafilters and Dedekind cuts of nonstandard arithmetic" (2014). *Electronic Theses and Dissertations*. 661.

http://digitalcommons.du.edu/etd/661

#### Provenance

Recieved from ProQuest

#### Rights holder

Timothy Onofre Trujillo

#### File size

181 p.

#### Copyright date

2014

#### File format

application/pdf

#### Language

en

#### Discipline

Mathematics