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1. Equivalence of codes for countable sets of reals

   https://doi.org/10.4153/S0008439520000661  

   Chan, William (September 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract A set $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ is universal for countable subsets of $${\mathbb {R}}$$ if and only if for all $$x \in {\mathbb {R}}$$ , the section $$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$$ is countable and for all countable sets $$A \subseteq {\mathbb {R}}$$ , there is an $$x \in {\mathbb {R}}$$ so that $$U_x = A$$ . Define the equivalence relation $$E_U$$ on $${\mathbb {R}}$$ by $$x_0 \ E_U \ x_1$$ if and only if $$U_{x_0} = U_{x_1}$$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $$E_{U^*}$$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $$E_U$$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $$E_U$$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $$\mathbf {\Sigma }_3^1$$ -generic absoluteness, then for all $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ that are $$\mathbf {\Sigma }_1^1$$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $$E_U$$ . 

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2. MOST(?) THEORIES HAVE BOREL COMPLETE REDUCTS

   https://doi.org/10.1017/jsl.2021.78  

   LASKOWSKI, MICHAEL C.; ULRICH, DOUGLAS S. (March 2023, The Journal of Symbolic Logic)

   Abstract We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete one-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $$M^{eq}$$ has a Borel complete reduct, and if a theory T is not $$\omega $$ -stable, then the elementary diagram of some countable model of T has a Borel complete reduct. 

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3. Equitable colourings of Borel graphs

   https://doi.org/10.1017/fmp.2021.12  

   Bernshteyn, Anton; Conley, Clinton T. (January 2021, Forum of Mathematics, Pi)

   Abstract Hajnal and Szemerédi proved that if G is a finite graph with maximum degree $$\Delta $$ , then for every integer $$k \geq \Delta +1$$ , G has a proper colouring with k colours in which every two colour classes differ in size at most by $$1$$ ; such colourings are called equitable. We obtain an analogue of this result for infinite graphs in the Borel setting. Specifically, we show that if G is an aperiodic Borel graph of finite maximum degree $$\Delta $$ , then for each $$k \geq \Delta + 1$$ , G has a Borel proper k -colouring in which every two colour classes are related by an element of the Borel full semigroup of G . In particular, such colourings are equitable with respect to every G -invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $$\Delta $$ -colourings of graphs with small average degree. Namely, we prove that if $$\Delta \geq 3$$ , G does not contain a clique on $$\Delta + 1$$ vertices and $$\mu $$ is an atomless G -invariant probability measure such that the average degree of G with respect to $$\mu $$ is at most $$\Delta /5$$ , then G has a $$\mu $$ -equitable $$\Delta $$ -colouring. As steps toward the proof of this result, we establish measurable and list-colouring extensions of a strengthening of Brooks’ theorem due to Kostochka and Nakprasit. 

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4. Tree-like graphings, wallings, and median graphings of equivalence relations

   https://doi.org/10.1017/fms.2025.22  

   Chen, Ruiyuan; Poulin, Antoine; Tao, Ran; Tserunyan, Anush (March 2025, Forum of Mathematics, Sigma)

   We prove several results showing that every locally finite Borel graph whose large-scale geometry is ‘tree-like’ induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree, answering a question of Tucker-Drob. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call aproper walling. The proof is based on the Stone duality between proper wallings and median graphs (i.e., CAT(0) cube complexes). Finally, we strengthen the conclusion of treeability in these results to hyperfiniteness in the case where the original graph has one (selected) end per component, generalizing the same result for trees due to Dougherty–Jackson–Kechris. 

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5. Borel asymptotic dimension and hyperfinite equivalence relations

   https://doi.org/10.1215/00127094-2022-0100  

   Conley, Clinton T; Jackson, Steve C; Marks, Andrew S; Seward, Brandon M; Tucker-Drob, Robin D (November 2023, Duke Mathematical Journal)

   A long-standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of solvable groups including the Baumslag–Solitar group BS(1, 2) and the lamplighter group Z2 ≀ Z. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov’s notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Følner tilings, topological dynamics, and C∗-algebras. 

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