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Infinite dimensional Ellentuck spaces

Friday 28th August 2015 - 09:00 to 10:00
INI Seminar Room 1

Topological Ramsey spaces have proved essential to solving certain problems in Banach spaces, Graph Theory, Set Theory, and Topology. In recent years, they have provided a mechanism for investigating initial Rudin-Keisler and Tukey structures in the Stone-Cech compactification of the natural numbers. In work of Dobrinen and Todorcevic and work of Dobrinen, Mijares, Trujillo, certain partial orders which force ultrafilters with partition relations were found to be equivalent to some new classes of topological Ramsey spaces. These in turn were used to precisely investigate the Ramsey-theoretic properties and the Tukey and Rudin-Keisler structures of the associated ultrafilters.

The Ellentuck space is the quintessential example of a topological Ramsey space. It is closely connected with Mathias forcing and with $\mathcal{P}(\omega)/\mathrm{Fin}$ which forces a Ramsey ultrafilter. Building on work in [Blass/Dobrinen/Raghavan15] investigating the Tukey type of the ultrafilter forced by $\mathcal{P}(\omega\times\omega)/\mathrm{Fin}\otimes\mathrm{Fin}$, we found in [DobrinenJSL15] that the essential structure responsible for the forcing properties actually is a 2-dimensional version of the Ellentuck space.

In this talk, we will present work in [DobrinenIDE15] constructing a new class of topological Ramsey spaces which may be viewed as infinite dimensional Ellentuck spaces. We will then present the Ramsey-classification theorems for equivalence relations on fronts and some applications to their related $\sigma$-closed forcings and the Rudin-Keisler and Tukey structures below the generic ultrafilters.

  • [Blass/Dobrinen/Raghavan15] Andreas Blass, Natasha Dobrinen, and Dilip Raghavan, The next best thing to a p-point, Journal of Symbolic Logic 80, no.3, 866-900.
  • [DobrinenJSL15] Natasha Dobrinen, High dimensional Ellentuck spaces and initial chains in the Tukey structure of non-p-points, Journal of Symbolic Logic, 27pp, To appear.
  • [DobrinenIDE15] Natasha Dobrinen, Infinite dimensional Ellentuck spaces, 35 + pp, Preprint.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons