We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin’s ultrafilter on the Turing degrees.

We define filter-tolerant powers of finite relational structures, and consider the problem of retracting such structures to their diagonal. We show that for some structures, such retractions can be readily derived from the axioms of Zermelo and Fraenkel, while at the other extreme, there are structures for which the existence of such retractions is equivalent to the ultrafilter axiom.

We show that, consistently, there is an ultrafilter F on ω such that if N n = (P l n ∪ Q l n, P l n, Q l n, R l n) (for l = 1, 2, n < ω), P l n ∪ Qn ⊆ ω, and Q

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk’s method more flexible, obtaining, for every cardinal λ of uncountable cofinality, a consistent example of a Boolean algebra Aλ whose every infinite homomorphic image is of cardin...

Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more uses multiple areas of mathematics, such as combinatorics, dynamics, algebra, among others. The purpose this article is to introduce ultrafilters friendly manner present some applications the branch combinatorics known Ramsey theory, culminating with new ultrafilter-based proof van der ...

We present some Zermelo-Fraenkel consistency results regarding bi-orderability of groups, as well a construction groups with Conradian orders whose every action on metric spaces has bounded orbits. A classical consequence the ultrafilter lemma is that group bi-orderable if and only it locally bi-orderable. show there exists model ZF in which free (ergo bi-orderable) not bi-orderable, can be giv...

We present a characterization of supercompactness measures for ω1 in L(R), and of countable products of such measures, using inner models. We give two applications of this characterization, the first obtaining the consistency of δ13 = ω2 with ZFC+AD , and the second proving the uniqueness of the supercompactness measure over Pω1 (λ) in L(R) for λ > δ21. Starting with the work of Steel [13] it b...

Hindman’s Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.