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Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

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Abstract

This paper investigates properties of \(\sigma \)-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter \({\mathcal {U}}\) for n-tuples, denoted \(t({\mathcal {U}},n)\), is the smallest number t such that given any \(l\ge 2\) and coloring \(c:[\omega ]^n\rightarrow l\), there is a member \(X\in {\mathcal {U}}\) such that the restriction of c to \([X]^n\) has no more than t colors. Many well-known \(\sigma \)-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by \(\sigma \)-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings \({\mathcal {P}}(\omega ^k)/\mathrm {Fin}^{\otimes k}\). We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these \(\sigma \)-closed forcings and their relationships with the classical pseudointersection number \({\mathfrak {p}}\).

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Acknowledgements

Dobrinen was supported by National Science Foundation Grants DMS-14247 and DMS-1600781. Navarro Flores was supported by CONACYT, and Dobrinen’s National Science Foundation Grants DMS-14247 and DMS-1600781.

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, USA

    Natasha Dobrinen

  2. Unidad Morelia, Universidad Nacional Autónoma de México, Morelia, Mexico

    Sonia Navarro Flores

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  1. Natasha Dobrinen
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  2. Sonia Navarro Flores
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Correspondence to Sonia Navarro Flores.

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Dobrinen was supported by National Science Foundation Grants DMS-1600781 and DMS-1901753. Navarro Flores was supported by CONACYT, and Dobrinen’s National Science Foundation Grant DMS-1600781.

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Dobrinen, N., Navarro Flores, S. Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces. Arch. Math. Logic 61, 1053–1090 (2022). https://doi.org/10.1007/s00153-022-00823-9

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