Strong Turing Determinacy, or ${\mathrm {sTD}}$, is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$.(2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every (...) set of reals is measurable and has Baire property.(3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.(4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$ :(a)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $, where $\mathrm {Dim_H}$ is the Hausdorff dimension.(b)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $, where $\mathrm {Dim_P}$ is the packing dimension. (shrink)

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional (...) uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations. (shrink)

We show that ZF + DC + “all Turing invariant sets of reals have the perfect set property” implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.