Abstract
A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not \(k+1\)-arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \(\mathcal {P}(n)\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \([\omega ]^{<\omega }\). In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bartoszyński, S.T., Judah, H.: Set Theory on the Structure of the Real Line. A. K. Peters Ltd, Wellesley (1995)
Baumgartner, J.E., Taylor, A.D.: Partition theorems and ultrafilters. Trans. Am. Math. Soc. 241, 283–309 (1978)
Blass, A.: The Rudin-Keisler ordering of P-points. Trans. Am. Math. Soc. 179, 145–166 (1973)
Di Prisco, C., Mijares, J., Nieto, J.: Local Ramsey theory. An abstract approach, Mathematical Logic Quarterly, 13 pp (2017)
Dobrinen, N.: Continuous cofinal maps on ultrafilters, 25 pp. Results have been incorporated into a new extended work (2010)
Dobrinen, N.: Continuous and other finitely generated canonical cofinal maps on ultrafilters. arXiv:1505.00368 (2015), 41 pp, Submitted
Dobrinen, N.: Survey on the Tukey theory of ultrafilters. Zbornik Radova 17(25), 53–80 (2015)
Dobrinen, N.: High dimensional Ellentuck spaces and initial chains in the Tukey structure of non-p-points. J. Symb. Log. 81(1), 237–263 (2016)
Dobrinen, N.: Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems. J. Math. Log. 16(1), 1650003 (2016)
Dobrinen, N., Todorcevic, S.: Tukey types of ultrafilters. Ill. J. Math. 55(3), 907–951 (2011)
Dobrinen, N., Todorcevic, S.: A new class of Ramsey-classification Theorems and their applications in the Tukey theory of ultrafilters, Part 1. Trans. Am. Math. Soc. 366(3), 1659–1684 (2014)
Dobrinen, N., Todorcevic, S.: A new class of Ramsey-classification Theorems and their applications in the Tukey theory of ultrafilters, Part 2. Trans. Am. Math. Soc. 367(7), 4627–4659 (2015)
Ellentuck, E.: A new proof that analytic sets are Ramsey. J. Symb. Log. 39(1), 163–165 (1974)
Erdős, P., Rado, R.: A combinatorial theorem. J. Lond. Math. Soc. 25, 249–255 (1950)
Farah, I.: Semiselective coideals. Mathematika 45(1), 79–103 (1998)
Kechris, A.S., Pestov, V.G., Todorcevic, S.: Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal. 15(1), 106–189 (2005)
Laflamme, C.: Forcing with filters and complete combinatorics. Ann. Pure Appl. Log. 42, 125–163 (1989)
Mijares, J.G.: A notion of selective ultrafilter corresponding to topological Ramsey spaces. Math. Log. Q. 53(3), 255–267 (2007)
Mijares, J., Padilla, G.: A Ramsey space of infinite polyhedra and the infinite random polyhedron, 20 pp, Submitted (2012)
Mijares, J., Torrealba, D.: A topological Ramsey space of metric structures. Acta Cientifica Venezolana (2017, accepted)
Nešetřil, J., Rödl, V.: Ramsey classes of set systems. J. Comb. Theory Ser. A 34, 183–201 (1983)
Pröml, H.J., Voigt, B.: Canonical forms of Borel-measurable mappings \({\Delta }: [\omega ]^{\omega }\rightarrow \mathbb{R}\). J. Comb. Theory Ser. A 40, 409–417 (1985)
Pudlák, P., Rödl, V.: Partition theorems for systems of finite subsets of integers. Discrete Math. 39, 67–73 (1982)
Raghavan, D., Todorcevic, S.: Cofinal types of ultrafilters. Ann. Pure Appl. Log. 163(3), 185–199 (2012)
Sokić, M.: Ramsey property of posets and related structures. Ph.D. thesis, University of Toronto (2010)
Todorcevic, S.: Introduction to Ramsey Spaces. Princeton University Press, Princeton (2010)
Trujillo, T.: Topological Ramsey spaces, associated ultrafilters, and their applications to the Tukey theory of ultrafilters and Dedekind cuts of nonstandard arithmetic. Ph.D. thesis, University of Denver (2014)
Trujillo, T.: Selective but not Ramsey. Topol. Appl. 202, 61–69 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to James Baumgartner, whose depth and insight continue to inspire.
Dobrinen was partially supported by National Science Foundation Grant DMS-1301665 and Simons Foundation Collaboration Grant 245286. Mijares was partially supported Dobrinen’s Simons Foundation Collaboration Grant 245286.
Rights and permissions
About this article
Cite this article
Dobrinen, N., Mijares, J.G. & Trujillo, T. Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points. Arch. Math. Logic 56, 733–782 (2017). https://doi.org/10.1007/s00153-017-0540-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-017-0540-0