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Transactions of the American Mathematical Society

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

Authors: Natasha Dobrinen and Stevo Todorcevic
Journal: Trans. Amer. Math. Soc. 367 (2015), 4627-4659
MSC (2010): Primary 05D10, 03E02, 06A06, 54D80; Secondary 03E04, 03E05
Published electronically: December 11, 2014
MathSciNet review: 3335396
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Abstract: Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces $ \mathcal {R}_{\alpha }$, $ \alpha <\omega _1$. These spaces form a natural hierarchy of complexity, $ \mathcal {R}_0$ being the Ellentuck space, and for each $ \alpha <\omega _1$, $ \mathcal {R}_{\alpha +1}$ coming immediately after $ \mathcal {R}_{\alpha }$ in complexity. Associated with each $ \mathcal {R}_{\alpha }$ is an ultrafilter $ \mathcal {U}_{\alpha }$, which is Ramsey for $ \mathcal {R}_{\alpha }$, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on $ \mathcal {R}_{\alpha }$, $ 2\le \alpha <\omega _1$. These form a hierarchy of extensions of the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to $ \mathcal {U}_{\alpha }$, for each $ 2\le \alpha <\omega _1$: Every nonprincipal ultrafilter which is Tukey reducible to $ \mathcal {U}_{\alpha }$ is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to $ \mathcal {U}_{\alpha }$ form a descending chain of rapid p-points of order type $ \alpha +1$.

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Additional Information

Natasha Dobrinen
Affiliation: Department of Mathematics, University of Denver, 2360 S. Gaylord Street, Denver, Colorado 80208

Stevo Todorcevic
Affiliation: Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4 – and – Institut de Mathematiques de Jussieu, CNRS - UMR 7056, 75205 Paris, France

Received by editor(s): June 13, 2012
Received by editor(s) in revised form: February 12, 2013
Published electronically: December 11, 2014
Additional Notes: The first author was supported by a National Science Foundation - Association for Women in Mathematics Mentoring Grant and a University of Denver Faculty Research Fund Grant
The second author was supported by grants from NSERC and CNRS
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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