|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|
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