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A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, Part 1


Authors: Natasha Dobrinen and Stevo Todorcevic
Journal: Trans. Amer. Math. Soc. 366 (2014), 1659-1684
MSC (2010): Primary 05D10, 03E02, 06A06, 54D80; Secondary 03E04, 03E05
DOI: https://doi.org/10.1090/S0002-9947-2013-05844-8
Published electronically: November 6, 2013
MathSciNet review: 3145746
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Abstract: Motivated by a Tukey classification problem, we develop a new topological Ramsey space $ \mathcal {R}_1$ that in its complexity comes immediately after the classical Ellentuck space. Associated with $ \mathcal {R}_1$ is an ultrafilter $ \mathcal {U}_1$ which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on $ \mathcal {R}_1$. This is analogous to the Pudlak-RödlTheorem 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 $ \mathcal {U}_1$: Every ultrafilter which is Tukey reducible to $ \mathcal {U}_1$ 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 $ \mathcal {U}_1$, namely the Tukey type of a Ramsey ultrafilter.


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Additional Information

Natasha Dobrinen
Affiliation: Department of Mathematics, University of Denver, 2360 S Gaylord Street, Denver, Colorado 80208
Email: natasha.dobrinen@du.edu

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
Email: stevo@math.toronto.edu; stevo@math.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9947-2013-05844-8
Received by editor(s): November 23, 2011
Received by editor(s) in revised form: March 31, 2012
Published electronically: November 6, 2013
Additional Notes: The first author was supported by an Association for Women in Mathematics - National Science Foundation Mentoring Travel Grant and a University of Denver Faculty Research Fund Grant
The second author was supported by grants from NSERC and CNRS
Article copyright: © Copyright 2013 American Mathematical Society

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