54 research outputs found
Topological Ramsey spaces from Fra\"iss\'e classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points
A general method for constructing a new class of topological Ramsey spaces is
presented. Members of such spaces are infinite sequences of products of
Fra\"iss\'e classes of finite relational structures satisfying the Ramsey
property. The Product Ramsey Theorem of Soki\v{c} is extended to equivalence
relations for finite products of structures from Fra\"iss\'e 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 Pudl\'ak-R\"odl
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
Fra\"iss\'e classes, we construct topological Ramsey spaces which are dense in
the partial orders of Baumgartner and Taylor in \cite{Baumgartner/Taylor78}
generating p-points which are -arrow but not -arrow, and in a partial
order of Blass in \cite{Blass73} producing a diamond shape in the Rudin-Keisler
structure of p-points. Any space in our framework in which blocks are products
of many structures produces ultrafilters with initial Tukey structure
exactly the Boolean algebra . If the number of Fra\"iss\'e
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
. In contrast, the set of isomorphism types of any product
of finitely many Fra\"iss\'e 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.Comment: 35 pages. Abstract and introduction re-written to make very clear the
main points of the paper. Some typos and a few minor errors have been fixe
Tukey types of ultrafilters
We investigate the structure of the Tukey types of ultrafilters on countable
sets partially ordered by reverse inclusion. A canonization of cofinal maps
from a p-point into another ultrafilter is obtained. This is used in particular
to study the Tukey types of p-points and selective ultrafilters. Results fall
into three main categories: comparison to a basis element for selective
ultrafilters, embeddings of chains and antichains into the Tukey types, and
Tukey types generated by block-basic ultrafilters on FIN.Comment: 33 pages, to appear in 2012 in the Illinois Journal of Mathematic
Quasi-selective ultrafilters and asymptotic numerosities
We isolate a new class of ultrafilters on N, called "quasi-selective" because
they are intermediate between selective ultrafilters and P-points. (Under the
Continuum Hypothesis these three classes are distinct.) The existence of
quasi-selective ultrafilters is equivalent to the existence of "asymptotic
numerosities" for all sets of tuples of natural numbers. Such numerosities are
hypernatural numbers that generalize finite cardinalities to countable point
sets. Most notably, they maintain the structure of ordered semiring, and, in a
precise sense, they allow for a natural extension of asymptotic density to all
sequences of tuples of natural numbers.Comment: 27 page
- …