2014/01/31 by Natasha Dobrinen, Dobrinen, Natasha, José G. Mijares +3
Computer Science · Mathematics · #03E02 #03E05 #03E40 #05C55 #05D10 #54H05 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1401.8105
openalex publication_date 2014/01/31 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
A general method for constructing a new class of topological Ramsey spaces is\npresented. Members of such spaces are infinite sequences of products of\nFra "iss 'e classes of finite relational structures satisfying the Ramsey\nproperty. The Product Ramsey Theorem of Soki vc is extended to equivalence\nrelations for finite products of structures from Fra "iss 'e classes of finite\nrelational structures satisfying the Ramsey property and the Order-Prescribed\nFree Amalgamation Property. This is essential to proving Ramsey-classification\ntheorems for equivalence relations on fronts, generalizing the Pudl 'ak-R "odl\nTheorem to this class of topological Ramsey spaces.\n To each topological Ramsey space in this framework corresponds an associated\nultrafilter satisfying some weak partition property. By using the correct\nFra "iss 'e classes, we construct topological Ramsey spaces which are dense in\nthe partial orders of Baumgartner and Taylor in citeBaumgartner/Taylor78\ngenerating p-points which are k-arrow but not k+1-arrow, and in a partial\norder of Blass in citeBlass73 producing a diamond shape in the Rudin-Keisler\nstructure of p-points. Any space in our framework in which blocks are products\nof n many structures produces ultrafilters with initial Tukey structure\nexactly the Boolean algebra \P(n). If the number of Fra "iss 'e\nclasses on each block grows without bound, then the Tukey types of the p-points\nbelow the space's associated ultrafilter have the structure exactly\n[\ω]<\ω. In contrast, the set of isomorphism types of any product\nof finitely many Fra "iss 'e classes of finite relational structures satisfying\nthe Ramsey property and the OPFAP, partially ordered by embedding, is realized\nas the initial Rudin-Keisler structure of some p-point generated by a space\nconstructed from our template.\n