Baire measurable paradoxical decompositions via matchings

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in $\mathbb{R}^3$ using Baire measurable pieces. We also obtain a Baire category solution to the dynamical von Neumann-Day problem: if $a$ is a nonamenable action of a group on a Polish space $X$ by Borel automorphisms, then there is a free Baire measurable action of $\mathbb{F}_2$ on $X$ which is Lipschitz with respect to $a$.Comment: Minor revision

Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research.Comment: Corrected typo

Scott Ranks of Classifications of the Admissibility Equivalence Relation

Let $\mathscr{L}$ be a recursive language. Let $S(\mathscr{L})$ be the set of $\mathscr{L}$-structures with domain $\omega$. Let $\Phi : {}^\omega 2 \rightarrow S(\mathscr{L})$ be a $\Delta_1^1$ function with the property that for all $x,y \in {}^\omega 2$, $\omega_1^x = \omega_1^y$ if and only if $\Phi(x) \approx_{\mathscr{L}} \Phi(y)$. Then there is some $x \in {}^\omega 2$ so that $\mathrm{SR}(\Phi(x)) = \omega_1^x + 1$