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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, Part 1
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by Natasha Dobrinen and Stevo Todorcevic PDF
Trans. Amer. Math. Soc. 366 (2014), 1659-1684 Request permission

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-Rödl 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 $\mathcal {U}_1$: 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
  • MR Author ID: 172980
  • Email: stevo@math.toronto.edu; stevo@math.jussieu.fr
  • 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
  • © Copyright 2013 American Mathematical Society
  • 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
  • MathSciNet review: 3145746