Authors: Dobrinen, Natasha
Todorčević, Stevo 
Title: A new class of Ramsey-Classification Theorems and their Applications in the Tukey Theory of Ultrafilters, Parts 1 and 2
Journal: Electronic Notes in Discrete Mathematics
Volume: 43
First page: 107
Last page: 112
Issue Date: 9-Sep-2013
ISSN: 1571-0653
DOI: 10.1016/j.endm.2013.07.018
Abstract: 
Motivated by Tukey classification problems, we develop a new hierarchy of topological Ramsey spaces Rα,α<ω1. These spaces form a natural hierarchy of complexity, R0 being the Ellentuck space [Erik Ellentuck, A new proof that analytic sets are Ramsey, Journal of Symbolic Logic 39 (1974), 163-165], and for each α<ω1,Rα+1 coming immediately after Rα in complexity. Associated with each Rα is an ultrafilter Uα, which is Ramsey for Rα, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on Rα,1≤α<ω1. 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 Uα, for each 1≤α<ω1: Every nonprincipal ultrafilter which is Tukey reducible to Uα 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 Uα form a descending chain of rapid p-points of order type α+1.
Keywords: Barrier | Erdos-Rado Theorem | Pudlák-Rödl Theorem | Ramsey-classification theorem | Tukey types | Ultrafilter
Publisher: Elsevier

Show full item record

Page view(s)

21
checked on Jan 30, 2023

Google ScholarTM

Check

Altmetric

Altmetric


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.