## 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.## References

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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