38 research outputs found
On the Tukey types of Fubini products
No full textWe extend the class of ultrafilters over countable sets for which , extending several results from \cite{Dobrinen/Todorcevic11}. In particular, we prove that for each countable ordinal , the generic ultrafilter forced by satisfy . This answers a question posed in \cite[Question 43]{Dobrinen/Todorcevic11}. Additionally, we establish that Milliken-Taylor ultrafilters possess the property that .accepted versio
FORCING IN RAMSEY THEORY (Infinite Combinatorics and Forcing Theory)
No full textRamsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on forcings which contain dense subsets forming topological Rarnsey spaces. These forcings motivated the development of new Ramsey theory, which then was applied to the generic ultrafilters to obtain the precise structure Rudin-Keisler and Tukey orders below such ultrafilters. The content of the first two tutorials has appeared in a previous paper [5]. The third tutorial concentrated on uses of forcing to prove Ramsey theorems for trees which are applied to determine big Ramsey degrees of homogeneous relational structures. This is the focus of this paper
Borel sets of Rado graphs and Ramsey\u27s Theorem
No full textThe well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset , where is endowed with the metric topology, each infinite subset contains an infinite subset such that is either contained in or disjoint from . Kechris, Pestov, and Todorcevic point out in their seminal 2005 paper the dearth of similar results for homogeneous structures. Such results are a necessary step to the larger goal of finding a correspondence between structures with infinite dimensional Ramsey properties and topological dynamics, extending their correspondence between the Ramsey property and extreme amenability. In this article, we prove an analogue of the Galvin-Prikry theorem for the Rado graph. Any such infinite dimensional Ramsey theorem is subject to constraints following from the 2006 work of Laflamme, Sauer, and Vuksanovic. The proof uses techniques developed for the author\u27s work on the Ramsey theory of the Henson graphs as well as some new methods for fusion sequences, used to bypass the lack of a certain amalgamation property enjoyed by the Baire space.Glitch in proof of Theorem 5.4 fixed, applying methods from arXiv:2203.00169. To appear in the European Journal of Combinatorics special issue for the Prague 2016 Ramsey Theory DocCours
Open Problems in the Reverse Mathematics of Ramsey Theory on Trees and Graphs
No full textNon UBCUnreviewedAuthor affiliation: University of DenverFacult
Topological Ramsey spaces in some creature forcings
No full textNon UBCUnreviewedAuthor affiliation: University of DenverFacult
Ramsey theory of the Henson graphs
No full textWe use techniques of logic, particularly set theory, to determine upper bounds on the big Ramsey degrees of the universal homogeneous k-clique-free graphs, for each k greater than two.Non UBCUnreviewedAuthor affiliation: University of DenverFacult
Survey on the Tukey theory of ultrafilters
No full textThis article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell’s Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guar-antee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made
The Ramsey Theory of Henson graphs
No full textAnalogues of Ramsey's Theorem for infinite structures such as the rationals
or the Rado graph have been known for some time. In this context, one looks for
optimal bounds, called degrees, for the number of colors in an isomorphic
substructure rather than one color, as that is often impossible. Such theorems
for Henson graphs however remained elusive, due to lack of techniques for
handling forbidden cliques. Building on the author's recent result for the
triangle-free Henson graph, we prove that for each , the
-clique-free Henson graph has finite big Ramsey degrees, the appropriate
analogue of Ramsey's Theorem.
We develop a method for coding copies of Henson graphs into a new class of
trees, called strong coding trees, and prove Ramsey theorems for these trees
which are applied to deduce finite big Ramsey degrees. The approach here
provides a general methodology opening further study of big Ramsey degrees for
ultrahomogeneous structures. The results have bearing on topological dynamics
via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 78 pages. To appear in Journal of Mathematical Logi
