Roberto Pinciroli
We consider countable Borel equivalence relations on quotient Borel spaces. We prove a generalization of the Feldman-Moore representation theorem, but provide some examples showing that other very simple properties of countable equivalence relations on standard Borel spaces may fail in the context of nonsmooth quotients.
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Finite-index extensions of essentially free countable Borel equivalence relations
本研究针对Kechris提出的可数Borel等价关系本质自由性的有限指数扩张保持性问题，证明E本质自由时其对应的E^fin也本质自由，从而给出该问题的肯定答案。
A dichotomy for countable unions of smooth Borel equivalence relations
本文证明了光滑 Borel 等价关系的可数并集满足二分性：要么可 Borel 归约到可数 Borel 等价关系，要么连续嵌入 $\mathbb{E}_1$，并推广到更一般情形。
We observe that Hall's free projective extension $P \mapsto F(P)$ of partial planes is a Borel map, and use a modification of the construction introduced in [9] to conclude that the class of countable non-Desarguesian projective planes is Borel complete. In the process, we also rediscover the main result of [7] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.
We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${\bfΣ}^0_1 \!\times\! {\bfΣ}^0_ξ$ sets, or by a ${\bfΠ}^0_1 \!\times\! {\bfΠ}^0_ξ$ set.
Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting …
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…
We prove that in some cases definable chains of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic chains, ROD chains in the Solovay model, and $Σ^1_2$ chains in the assumption that $ω_1^{L[x]}<ω_1$ for all reals $x$.
We study Borel equivalence relations induced by Borel actions of tsi Polish groups on standard Borel spaces. We characterize when such an equivalence relation admits classification by countable structures using a variant of the $\mathbb G_0$-dichotomy. In particular, we find a class that serves as a base for non-classification by countable structures for these equivalence relations under Borel reducibility. We use this characterization together with the result of [B. D. Miller, to appear in the Journal of Mathematical Logic] to show that if such an equivalence relation admits classification by…
No Countable Basis for Borel Directed Graphs of Dichromatic Number at Least Three
本文证明了Borel有向图中顶点可被分成两个Borel无环集的图类为Σ¹₂完全集，表明判断Borel有向图的二色数至少为三是一个Π¹₂完全问题。
In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel quasi-orders. In this paper we are concerned with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We…
The Borel Ramsey properties for countable Borel equivalence relations
本文定义了可数Borel等价关系的强弱Borel Ramsey性质，证明其成立当且仅当关系光滑，并探讨了超有限非光滑情形的变体。
Hausdorff dimension and countable Borel equivalence relations
本文证明可数 Borel 等价关系可在满 Hausdorff 维数的闭集上光滑化，并将结论推广到局部可数 Borel 拟序与低阶尺度函数情形。
Let $E\subseteq F$ and $E'\subseteq F'$ be Borel equivalence relations on the standard Borel spaces $X$ and $Y$, respectively. The pair $(E,F)$ is simultaneously Borel reducible to the pair $(E',F')$ if there is a Borel function $f:X\to Y$ that is both a reduction from $E$ to $E'$ and a reduction from $F$ to $F'$. Simultaneous Borel embeddings and isomorphisms are defined analogously. We classify all pairs $E\subseteq F$ of smooth countable Borel equivalence relations up to simultaneous Borel bireducibility and biembeddability, and a significant portion of such pairs up to simultaneous Borel i…
Countable Borel treeable equivalence relations are classifiable by $\ell_1$
该研究通过自由巴拿赫空间归约证明可树化可数Borel等价关系可由阿贝尔波兰群分类，否定了Hjorth的相关猜想，同时证明可由局部紧阿贝尔波兰群可数积分类的CBER是超有限的，用到Hjorth波兰群作用分析方法。
This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy. We show that if $E$ is strongly ergodic with respect to $μ$ then $E^\mathbb{N}$ is strongly ergodic with respect to $μ^\mathbb{N}$. We answer questions of Clemens and Coskey regarding their recently defined $Γ$-jump operations, in particular showing that the $\mathbb{Z}^2$-jump of $E_\infty$ is strictly above the $\mathbb{Z}$-jump of $E_\infty$. We study a notion of equivalence relations which can be classified by infinite sequences…
We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.
Definable expansions on countable groups and countable Borel equivalence relations
本文在描述组合学框架下研究可数 Borel 等价关系与可数群的可定义扩张，建立二者对应及测度、范畴结构定理，并通过生成树、Ramsey 单色集和偏序线性化等示例说明。
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence relations imposed by the notion of Borel reducibility. We will survey results and conjectures concerning the problem of identifying the place in this hierarchy of these equivalence relations from…
We give, for each countable ordinal $ξ\geq 1$, an example of a ${\bfΔ}^0_2$ countable union of Borel rectangles that cannot be decomposed into countably many ${\bfΠ}^0_ξ$ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ${\bfΔ}^0_ξ$-measurable countable coloring.
Borel completeness of the class of countable Steiner triple systems
该研究证明可数施泰纳三元系的同构关系是博雷尔完备的，通过构造保持自同构的博雷尔归约完成证明，拓展了博雷尔完备性在组合结构中的相关结论。
Given a countable o-minimal theory T, we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T. This is done by constructing models with specific linear orders in the tail of the Archimedean ladder of a suitable nonsimple type. If the theory admits no nonsimple types, then we use Mayer's characterization of isomorphism for such theories to compute invariants for countabl…
We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-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 $ω$-stable, then the elementary diagram of some countable model of $T$ has a Borel complete reduct.
We study topological groups that can be defined as Polish, pro-countable abelian groups, as non-archimedean abelian groups or as quasi-countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups such that all orbit equivalence relations induced by their continuous actions on Polish spaces are Borel, and relatively tame groups $G$, i.e., groups such that every diagonal action $α\times β$ of $G$ induces a Borel orbit equivalence relation, provided that the actions $α$, $β$ of $…
In this paper we show that it is relatively consistent with ZFC that every gamma-set is countable while not every strong measure zero set is countable. This answers a question of Paul Szeptycki. A set is a gamma-set iff every omega-cover contains a gamma-subcover. An open cover is an omega-cover iff every finite set is covered by some element of the cover. An open cover is a gamma-cover iff every element of the space is in all but finitely many elements of the cover. Gerlits and Nagy proved that every gamma-set has strong measure zero. We also show that is consistent that every strong gamma-se…
In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we prove that if the real line is the countable union of countable sets, then there exists an F-sigma-delta set which is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite set of reals which is F-sigma-delta.
We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.
Let $F_{ω_1}$ be the countable admissible ordinal equivalence relation defined on ${}^ω2$ by $x \ F_{ω_1} \ y$ if and only if $ω_1^x = ω_1^y$. It will be shown that $F_{ω_1}$ is classifiable by countable structures and must be classified by structures of high Scott rank. If $E$ and $F$ are equivalence relations, then $E$ is almost Borel reducible to $F$ if and only if there is a Borel reduction of $E$ to $F$, except possibly on countably many $E$-classes. Let $E_{ω_1}$ denote the equivalence of order types of reals coding well-orderings. It will be shown that in the constructible universe $L$ …
Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order
本文证明了无 torsion 的 Abel 群是忠实 Borel 完整的，并展示了纯可嵌入性是一个完全的解析准序。
Decompositions and measures on countable Borel equivalence relations
本文证明可数Borel等价关系的测度论遍历分解可由生成它的群作用的拓扑遍历分解实现，并应用于基数代数与弱等价性研究。
We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating $μ$-hyperfinite equivalence relations admit $μ$-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a c…
In this article, we give a full description of the Wadge degrees of Borel functions from $ω^ω$ to a better quasi ordering $\mathcal{Q}$. More precisely, for any countable ordinal $ξ$, we show that the Wadge degrees of $\mathbfΔ^0_{1+ξ}$-measurable functions $ω^ω\to\mathcal{Q}$ can be represented by countable joins of the $ξ$-th transfinite nests of $\mathcal{Q}$-labeled well-founded trees.
For a class $\mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there is a Borel way to put a structure in $\mathcal K$ on each $E$-equivalence class. We study in this paper the global structure of the classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$. We show that $\mathcal K$-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of $\…
Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{ω_1ω}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{ω_1ω}(Q)$-definable sets.
In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that a strong form of locally countable graphs with compact totally disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model in the model obtained by adding $\aleph_2$ Laver reals. From this, we answer a question from Geschke and the second author (2022), and another question from Fisher, Friedman and Khomskii (2014) concerning regularity properties of subsets of the real line.
We prove a number of results motivated by global questions of uniformity in computability 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 …
We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise…
We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order.
Borel completeness of Tits buildings with no rank 3 residues of spherical type
本文研究了无球型3级残余的蒂茨建筑的博雷尔完备性，证明了其分类难度与可数图分类相当，并特别指出广义n边形的博雷尔完备性。
We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB, the class of torsion-free abelian groups. We also prove that for any countable ring $R$, both the class of left $R$-modules endowed with an endomorphism and the class of left $R$-modules with four named submodules are Borel complete.
Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.
We show that for any infinite countable group $G$ and for any free Borel action $G \curvearrowright X$ there exists an equivariant class-bijective Borel map from $X$ to the free part $\mathrm{Free}(2^G)$ of the $2$-shift $G \curvearrowright 2^G$. This implies that any Borel structurability which holds for the equivalence relation generated by $G \curvearrowright \mathrm{Free}(2^G)$ must hold a fortiori for all equivalence relations coming from free Borel actions of $G$. A related consequence is that the Borel chromatic number of $\mathrm{Free}(2^G)$ is the maximum among Borel chromatic numbers…
We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $Φ$ as a class of structures in a related language. From this, we show that $Φ$ has a Borel complete expansion if and only if $S_\infty$ divides $Aut(M)$ for some countable model $M\models Φ$. Using this, we prove that for theories $T_h$ asserting that $\{E_n\}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of \cite{LU}.
We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbfΣ^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth--Kechris about discrete structures. As a different application, we also give a new proof of Kechr…
Isomorphism of locally compact Polish metric structures
本文研究局部紧 Polish 度量结构 Borel 类上的同构，证明其可由可数结构分类，并给出描述复杂度归约及博弈论不可分类判据。
Reductions and necessary conditions for tall Borel Ramsey ideals
研究高博雷尔理想是否具拉姆齐性质这一开放问题，通过主要定理给出不存在可数局部解读等结论，还证明弱选择性$q^+$理想相关性质，提出博雷尔性是否强制可数解读的问题及后果，主要问题仍待解决。
In this paper we study the Borel structure of the space of left-orderings $\mathrm{LO}(G)$ of a group $G$ modulo the natural conjugacy action, and by using tools from descriptive set theory we find many examples of countable left-orderable groups such that the quotient space $\mathrm{LO}(G)/G$ is not standard. This answers a question of Deroin, Navas, and Rivas. We also prove that the countable Borel equivalence relation induced from the conjugacy action of $\mathbb{F}_{2}$ on $\mathrm{LO}(\mathbb{F}_{2})$ is universal, and leverage this result to provide many other examples of countable left-…
We consider the conjugacy problem for the automorphism groups of a number of countable homogeneous structures. In each case we find the precise complexity of the conjugacy relation in the sense of Borel reducibility.
Realizations of countable Borel equivalence relations
本文研究可数Borel等价关系的拓扑实现，涵盖连续群作用、极小子移位及普适空间中的性质刻画，并探讨Turing等价等特殊关系的实现问题。
It is shown that Borel games of length $ω^2$ are determined if, and only if, for every countable ordinal $α$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $α$-many iterated powersets above a limit of Woodin cardinals.
We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic $\mathcal L_{ω_1ω}$: every countable $\mathcal L_{ω_1ω}$-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories $(\mathcal L, \mathcal T)$ and $(\mathcal L', \mathcal T')$ (in possibly different languages $\mathcal L, \mathcal L'$), every Borel functor $\mathsf{Mod}(\mathcal L', \mathcal T') \to \mathsf{Mod}(\mathcal L, \mathcal T)$ between the respective groupoids of count…
Isomorphism of pointed minimal systems is not classifiable by countable structures
本文证明极小系统及带点极小系统的拓扑共轭关系无法通过可数结构进行 Borel 归约分类，揭示了其分类复杂性的本质。
Analogues of the countable Borel equivalence relations in the setting of computable reducibility
本文研究可计算归约下可数Borel等价关系的两种类似物，证明基于Feldman-Moore有效化的概念存在严格二分法，而基于Lusin-Novikov的概念则不存在，并存在介于相等与几乎相等关系之间的链和反链，从而否定了Glimm-Efros二分法的一般类似物。
We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of E1 into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.
Structurable equivalence relations and $\mathcal{L}_{ω_1ω}$ interpretations
本文建立了可数 Borel 等价关系范畴与可解释特定理论的可数 $\mathcal{L}_{ω_1ω}$ 理论范畴之间的对偶等价，将 Borel 组合结构问题转化为语法可定义性问题，并推广到局部可数 Borel 群胚。
Techniques of combinatorial set theory are applied to the following algebraic problem. Suppose G is an abelian group such that, for all countable subgroups C, the divisible part of the quotient G/C is countable. What can one conclude about the size of the divisible part of G/K when the cardinality of the subgroup K is a given uncountable cardinal?
We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of ZFC.
Borel Order Dimension
本文引入Borel拟序的Borel序维数，借助Borel二色数二分法刻画不可数维数并证明可数维数拟序可Borel线性化，还研究局部可数情形及Turing度。
Borel completeness of $R$-modules when $R$ fails the DCC on pp-definable subgroups
该研究证明了非左完全可数环的R-模理论是Borel完备的，刻画了可数单环的Borel完备性，引入了相关理想与包络构造，强化了TFAB的Borel完备性结论。
Equivalence of Random Generics Is Not Essentially Free
该研究针对ZFC模型上随机泛型的等价关系，结合Popa超刚性等工具证明其正测度限制不可弱归约到本质自由等价关系。
For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P := \{ gC : g \in G, C \in P \}$ generates the Borel $σ$-algebra of $X$. For $G = Z$, the Kolmogorov--Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite entropy. It was asked by Benjamin Weiss in the late 80s whether the nonexistence of any invariant probability measure guarantees the existence of a finite generator. We…
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