Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems

Journal of Mathematical Logic 16 (1):1650003 (2016)
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Abstract

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text].

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References found in this work

Happy Families.A. R. D. Mathias - 1977 - Annals of Mathematical Logic 12 (1):59.
A New Proof That Analytic Sets Are Ramsey.Erik Ellentuck - 1974 - Journal of Symbolic Logic 39 (1):163-165.
Every Analytic Set is Ramsey.Jack Silver - 1970 - Journal of Symbolic Logic 35 (1):60-64.
Borel Sets and Ramsey's Theorem.Fred Galvin & Karel Prikry - 1973 - Journal of Symbolic Logic 38 (2):193-198.

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