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Let Tq = -d2/dx2 +q be a Schr¨odinger operator in the space L2(R). A potential q is called reflectionless if the operator Tq is reflectionless. Let Q be the set of all reflectionless potentials of the Schr¨odinger operator, and let M be the set of nonnegative Borel measures on R with compact support. As shown by Marchenko, each potential q ∈ Q can be associated with a unique measure μ ∈ M. As a result, we get the bijection Θ: Q → M. In this paper, we show that one can define topologies on Q andM, under which the mapping Θ is a homeomorphism. [ABSTRACT FROM AUTHOR]

Descriptive set theory and computability theory are closely-related fields of logic; both are oriented around a notion of descriptive complexity. However, the two fields typically consider objects of very different sizes; computability theory is principally concerned with subsets of the naturals, while descriptive set theory is interested primarily in subsets of the reals. In this paper, we apply a generalization of computability theory, admissible recursion theory, to consider the relative complexity of notions that are of interest in descriptive set theory. In particular, we examine the perfect set property, determinacy, the Baire property, and Lebesgue measurability. We demonstrate that there is a separation of descriptive complexity between the perfect set property and determinacy for analytic sets of reals; we also show that the Baire property and Lebesgue measurability are both equivalent in complexity to the property of simply being a Borel set, for $\boldsymbol {\Sigma ^{1}_{2}}$ sets of reals. [ABSTRACT FROM AUTHOR]

Hjorth (Variations on Scott, 1998; The fine structure and Borel complexity of orbits, 2010) introduced a Scott analysis for general Polish group actions, and asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank—namely, if the orbit equivalence relation is Borel, then Hjorth ranks are bounded. We answer Hjorth's question positively. As a corollary we prove the following conjecture of Hjorth—for every limit ordinal α , the set of elements whose orbit is of complexity less than α is a Borel set. [ABSTRACT FROM AUTHOR]

In this paper, we study the Borel structure of the space of left‐orderings LO(G)$\operatorname{LO}(G)$ of a group G$G$ 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 LO(G)/G$\operatorname{LO}(G)/G$ 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 F2$\mathbb {F}_{2}$ on LO(F2)$\operatorname{LO}(\mathbb {F}_{2})$ is universal, and leverage this result to provide many other examples of countable left‐orderable groups G$G$ such that the natural G$G$‐action on LO(G)$\operatorname{LO}(G)$ induces a universal countable Borel equivalence relation. [ABSTRACT FROM AUTHOR]

Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc.230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory T and the Borel rank of the isomorphism relation ≅ T κ on its models of size κ , for κ any cardinal satisfying κ < κ = κ > 2 ℵ 0 . This is achieved by establishing a link between said rank and the ℒ ∞ κ -Scott height of the κ -sized models of T , and yields to the following descriptive set-theoretical analog of Shelah's Main Gap Theorem: Given a countable complete first-order theory T , either ≅ T κ is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is κ + > ℵ 1 ), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah's theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of ≅ T κ , and provide a characterization of categoricity of T in terms of the descriptive set-theoretical complexity of ≅ T κ . [ABSTRACT FROM AUTHOR]

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. [ABSTRACT FROM AUTHOR]

We prove the Decomposability Conjecture for functions of Baire class 2 from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory. [ABSTRACT FROM AUTHOR]

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{emb} $ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\Delta _2^0 $ -degrees of the Scott domain. We then show that $\mathbb{P}_{emb} $ admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\Delta _2^0 $ -degrees of the Scott domain. [ABSTRACT FROM AUTHOR]

We show, assuming weak large cardinals, that in the context of games of length ω with moves coming from a proper class, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of L that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, seventh, or (ω + 2) th order arithmetic. We also prove bounds on the strength of Borel determinacy for proper class games. These results answer questions of Gitman and Hamkins. [ABSTRACT FROM AUTHOR]

We give new formulae for analytic balayage of measures supported on subsets of Carathéodory compact sets in the complex plane, and consider a related problem of description of badly approximable functions in L p ( T ) . [ABSTRACT FROM AUTHOR]

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set D of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of D . When D contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If D is well-ordered the situation is more complex: for isometry we have an increasing sequence of Borel equivalence relations of length ω 1 which are cofinal among Borel equivalence relations classifiable by countable structures, while for isometric embeddability we have an increasing sequence of analytic quasi-orders of length at least ω + 3 . We then apply our results to solve various open problems in the literature. For instance, we answer a long-standing question of Gao and Kechris by showing that the relation of isometry on locally compact ultrametric Polish spaces is Borel bireducible with countable graph isomorphism. [ABSTRACT FROM AUTHOR]

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring. [ABSTRACT FROM AUTHOR]

In terms of a group G of isometries of Euclidean space, it is given a necessary and sufficient condition for the uniqueness of a G-measure on the Borel σ-algebra of this space. [ABSTRACT FROM AUTHOR]

Some pathological subsets of the real line R are considered from the viewpoint of their potential definability and their measurability properties with respect to different classes of measures on R. [ABSTRACT FROM AUTHOR]

Abstract: We prove a recursion theoretic characterization of the Topological Vaught Conjecture in the Zermelo‐Fraenkel set theory by using tools from effective descriptive set theory and by revisiting the result of Miller that orbits in Polish G‐spaces are Borel sets. [ABSTRACT FROM AUTHOR]

We show that a radial continuous valuation defined on the n -dimensional star bodies extends uniquely to a continuous valuation on the n -dimensional bounded star sets. Moreover, we provide an integral representation of every such valuation, in terms of the radial function, which is valid on the dense subset of the simple Borel star sets. Along the way, we also show that every radial continuous valuation defined on the n -dimensional star bodies can be decomposed as a sum V = V + − V − , where both V + and V − are positive radial continuous valuations. [ABSTRACT FROM AUTHOR]

We investigate measure theoretic properties of regular sets of infinite trees. As a first result, we prove that every regular set is universally measurable and that every Borel measure on the Polish space of trees is continuous with respect to a natural transfinite stratification of regular sets into ω 1 ranks. We also expose a connection between regular sets and the σ -algebra of R -sets , introduced by A. Kolmogorov in 1928 as a foundation for measure theory. We show that the game tree languages W i , k are Wadge-complete for the finite levels of the hierarchy of R -sets. We apply these results to answer positively an open problem regarding the game interpretation of the probabilistic μ -calculus. [ABSTRACT FROM AUTHOR]

The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence relation. [ABSTRACT FROM AUTHOR]

For each a ∈ ω ω , we define a Baire class one function f a : ω ω → ω ω which encodes a in a certain sense. We show that for each Borel g : ω ω → ω ω , f a ∩ g = ∅ implies a ∈ Δ 1 1 ( c ) where c is any code for g . We generalize this theorem for g in a larger pointclass Γ. Specifically, when Γ = Δ 2 1 , a ∈ L [ c ] . Also for all n ∈ ω , when Γ = Δ 3 + n 1 , a ∈ M 1 + n ( c ) . [ABSTRACT FROM AUTHOR]

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 countable models. If the theory is small, then the invariant is real-valued, and therefore its isomorphism relation is smooth. If not, the invariant corresponds to a countable set of reals, and therefore the isomorphism relation is Borel equivalent to F2.Combining these two results, we conclude that $\left( {{\rm{Mod}}\left( T \right), \cong } \right)$ is either maximally complicated or maximally uncomplicated (subject to completely general model-theoretic lower bounds based on the number of types and the number of countable models). [ABSTRACT FROM AUTHOR]

We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form $\Sigma _{1 + \alpha + 3}^0 $, for α < ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α + 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of ${\rm{\Sigma }}_{1 + \alpha + 3}^0 $ determinacy, for $\alpha < \omega _1^{CK} $. This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: The least θ so that all winning strategies in ${\rm{\Sigma }}_4^0 $ games belong to Lθ+1 is the least so that $L_\theta \models {\rm{``}}{\cal P}\left( \omega \right)$ exists, and all wellfounded trees are ranked”. [ABSTRACT FROM AUTHOR]

It is well known how to generalize the meagre ideal replacing $$\aleph _0$$ by a (regular) cardinal $$\lambda > \aleph _0$$ and requiring the ideal to be $$({<}\lambda )$$ -complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing $$\aleph _0$$ by $$\lambda $$ . So naturally, to call it a generalization we require it to be $$({<}\lambda )$$ -complete and $$\lambda ^+$$ -c.c. and more. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of $$\lambda $$ -Borel sets modulo the ideal. Common wisdom have said that there is no such thing because we have no parallel of Lebesgue integral, but here surprisingly first we get a positive $$=$$ existence answer for a generalization of the null ideal for a 'mild' large cardinal $$\lambda $$ -a weakly compact one. Second, we try to show that this together with the meagre ideal (for $$\lambda $$ ) behaves as in the countable case. In particular, we consider the classical Cichoń diagram, which compares several cardinal characterizations of those ideals. We shall deal with other cardinals, and with more properties of related forcing notions in subsequent papers (Shelah in The null ideal for uncountable cardinals; Iterations adding no $$\lambda $$ -Cohen; Random $$\lambda $$ -reals for inaccessible continued; Creature iteration for inaccesibles. Preprint; Bounding forcing with chain conditions for uncountable cardinals) and Cohen and Shelah (On a parallel of random real forcing for inaccessible cardinals. [math.LO]) and a joint work with Baumhauer and Goldstern. [ABSTRACT FROM AUTHOR]

Kechris showed in Kechris (Trans. Am. Math. Soc. 202, 259-297, 1975) that there exists a largest ${\Pi ^{1}_{1}}$ set of measure 0. An explicit construction of this largest ${\Pi ^{1}_{1}}$ nullset has later been given in Hjorth and Nies (J. Lond. Math. Soc. 75(2), 495-508, 2007). Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned in Chong and Yu (J. Symb. Log. 80(04), 1131-1148, 2015) and Yu (Fundam. Math. 215, 219-231, 2011)). In this paper, we refute this conjecture and show that this nullset is merely ${\Sigma }^{0}_{3}$ . Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is ${\Sigma }^{0}_{3}$ . To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties. [ABSTRACT FROM AUTHOR]

We study integralgeometric representations of variations of general sets A ⊂ R n without any regularity assumptions. If we assume, for example, that just one partial derivative of its characteristic function χ A is a signed Borel measure on R n with finite total variation, can we provide a nice integralgeometric representation of this variation? This is a delicate question, as the Gauss–Green type theorems of De Giorgi and Federer are not available in this generality. We will show that a ‘measure-theoretic boundary’ plays its role in such representations similarly as for the sets of finite variation. There is a variety of suitable notions of ‘measure-theoretic boundary’ and one can address the question to find notions of measure-theoretic boundary that are as fine as possible. [ABSTRACT FROM AUTHOR]

We investigate the determinacy strength of infinite games whose winning sets are recognized by nondeterministic pushdown automata with various acceptance conditions, e.g., safety, reachability and co-B¨uchi conditions. In terms of the foundational program "Reverse Mathematics", the determinacy strength of such games is measured by the complexity of a winning strategy required by the determinacy. Infinite games recognized by nondeterministic pushdown automata have some resemblance to those by deterministic 2-stack visibly pushdown automata with the same acceptance conditions. So, we first investigate the determinacy of games recognized by deterministic 2-stack visibly pushdown automata, together with that by nondeterministic ones. Then, for instance, we prove that the determinacy of games recognized by pushdown automata with a reachability condition is equivalent to the weak K¨onig lemma, stating that every infinite binary tree has an infinite path. While the determinacy for pushdown ω-languages with a B¨uchi condition is known to be independent from ZFC, we here show that for the co-B¨uchi condition, the determinacy is exactly captured by ATR0, another popular system of reverse mathematics asserting the existence of a transfinite hierarchy produced by iterating arithmetical comprehension along a given well-order. Finally, we conclude that all results for pushdown automata in this paper indeed hold for 1-counter automata. [ABSTRACT FROM AUTHOR]

We survey current stage of effective descriptive set theory which was fast evolving in the last decade. Most attention will be given to (effective) quasi-Polish spaces and their important subclasses (Polish spaces, ω -continuous domains) but also some important non-countably-based spaces (like the Kleene-Kreisel functionals or spaces of polynomials) will be discussed. Along with effective DST, also classical DST will be considered, which is the "limit" of the effective DST relative to arbitrary oracles. We will discuss not only classical hierarchies of sets and functions but also versions of (effective) Wadge hierarchy where a considerable progress was recently achieved. [ABSTRACT FROM AUTHOR]

Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical characteristics, similar to those implied by the Axiom of Determinacy: if λ witnesses I0, then there is a topology for V λ + 1 that is completely metrizable and with weight λ (i.e., it is a λ -Polish space), and it turns out that all the subsets of V λ + 1 in L (V λ + 1) have the λ -Perfect Set Property in such topology. In this paper, we find another generalized Polish space of singular weight κ of cofinality ω such that all its subsets have the κ -Perfect Set Property, and in doing this, we are lowering the consistency strength of such property from I0 to κ θ -supercompact, with θ > κ inaccessible. [ABSTRACT FROM AUTHOR]

We show that the model for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture given in actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets. [ABSTRACT FROM AUTHOR]

We provide a list of open problems in the research area of generalised Baire spaces, compiled with the help of the participants of two workshops held in Amsterdam (2014) and Hamburg (2015). [ABSTRACT FROM AUTHOR]

The "visibility" of a planar set S from a point a is defined as the normalized size of the radial projection of S from a to the unit circle centered at a. Simon and Solomyak in 2006 proved that unrectifiable self-similar onesets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of δ-neighborhoods of such sets. We also prove lower bounds on the visibility of δ-neighborhoods of more general sets, based in part on Bourgain's discretized sum-product estimates. [ABSTRACT FROM AUTHOR]

Let μ h , μ g be Hausdorff measures on compact metric spaces X , Y and let B o r ( X ) / J σ ( μ h ) and B o r ( Y ) / J 0 ( μ g ) be the Boolean algebras of Borel sets modulo σ -ideals of Borel sets that can be covered by countably many compact sets of σ -finite μ h -measure or μ g -measure null, respectively. We shall show that if μ h is not σ -finite, and one of the quotient Boolean algebras embeds densely in the other, then for some Borel B with μ h ( B ) = ∞ , μ h takes on Borel subsets of B only values 0 or ∞. This is a particular instance of some more general results concerning Boolean algebras B o r ( X ) / J , where J is a σ -ideal of Borel sets generated by compact sets. [ABSTRACT FROM AUTHOR]

We begin the fine analysis of nonBorel pointclasses. Working in ZFC + DET$\left( {_1^1 } \right)$, we describe the Wadge hierarchy of the class of increasing differences of co-analytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets. [ABSTRACT FROM PUBLISHER]

We define a notion of reducibility for subsets of a second countable T topological space based on relatively continuous relations and admissible representations. This notion of reducibility induces a hierarchy that refines the Baire classes and the Hausdorff-Kuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces. However in virtually every second countable T space, it yields a hierarchy on Borel sets, namely it is well founded and antichains are of length at most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line $${\mathbb{R}}$$ or the Scott Domain $${\mathcal{P}\omega}$$ . [ABSTRACT FROM AUTHOR]

We prove results that falsify Silver’s dichotomy for Borel equivalence relations on the generalized Baire space under the assumption V = L. [ABSTRACT FROM PUBLISHER]

We consider an infinite two-player stochastic zero-sum game with a Borel winning set, in which the opponent's actions are monitored via stochastic private signals. We introduce two conditions of the signalling structure: Stochastic Eventual Perfect Monitoring (SEPM) and Weak Stochastic Eventual Perfect Monitoring (WSEPM). When signals are deterministic these two conditions coincide and by a recent result due to Shmaya (2011) entail determinacy of the game. We generalize Shmaya's (2011) result and show that in the stochastic learning environment SEPM implies determinacy while WSEPM does not. [ABSTRACT FROM AUTHOR]

We show that some set-theoretic assumptions (for example Martin's Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets (in ℝn). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the σ-algebra of sets with the property of Baire. [ABSTRACT FROM AUTHOR]

The article examines how the Banach-Tarski Paradox and the Sierpin'ski sets can be given concrete interpretations in the hyperbolic plane. It notes that a point in the plane can be selected for any discrete group of isometries of hyperbolic plane. The work of authors R. Dougherty and M. Foreman demonstrates that the Banach-Tarski paradox is possible using pieces having the property of Baire sets, the union between of a Borel set and a meager set.

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 first establish that there is a universal countable Borel quasi-order, and then establish that several countable Borel quasi-orders are universal. An important example is an embeddability relation on descriptive set theoretic trees. Our main result states that embeddability of finitely generated groups is a universal countable Borel quasi-order, answering a question of Louveau and Rosendal. This immediately implies that biembeddability of finitely generated groups is a universal countable Borel equivalence relation. The same techniques are also used to show that embeddability of countable groups is a universal analytic quasi-order. Finally, we show that, up to Borel bireducibility, there are 2ℵ0 distinct countable Borel quasi-orders, which symmetrize to a universal countable Borel equivalence relation. [ABSTRACT FROM AUTHOR]

Abstract: Whenever the defining sequence of a Carleman ultraholomorphic class (in the sense of H. Komatsu) is strongly regular and associated with a proximate order, flat functions are constructed in the class on sectors of optimal opening. As consequences, we obtain analogues of both Borel–Ritt–Gevrey theorem and Watson's lemma in this general situation. [Copyright &y& Elsevier]

Abstract: We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals. Our methods include the projection method introduced by the authors in an earlier work, as well as several new methods. Some special consequences of our main results are (definitions provided in the paper): [(1)] Every product of a concentrated space with a Hurewicz space satisfies . On the other hand, assuming the Continuum Hypothesis, for each Sierpiński set S there is a Luzin set L such that can be mapped onto the real line by a Borel function. [(2)] Assuming Semifilter Trichotomy, every concentrated space is productively Menger and productively Rothberger. [(3)] Every scale set is productively Hurewicz, productively Menger, productively Scheepers, and productively Gerlits–Nagy. [(4)] Assuming , every productively Lindelöf space is productively Hurewicz, productively Menger, and productively Scheepers. A notorious open problem asks whether the additivity of Rothberger's property may be strictly greater than , the additivity of the ideal of Lebesgue-null sets of reals. We obtain a positive answer, modulo the consistency of Semifilter Trichotomy ( ) with . Our results improve upon and unify a number of results, established earlier by many authors. [Copyright &y& Elsevier]

Abstract: We give, for each countable ordinal , an example of a countable union of Borel rectangles that cannot be decomposed into countably many 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 -measurable countable coloring. [Copyright &y& Elsevier]

There does not exist a Borel selection of an isomorphism class within each bi-embeddability class of finitely generated groups. [ABSTRACT FROM AUTHOR]

Let H be an infinite dimensional Hilbert space, X a compact Hausdorff space and f : X → X a homeomorphism which preserves a Borel ergodic probability measure which is positive on non-empty open sets. We prove that non-uniformly Anosov cocycles are C0-dense in the family of partially hyperbolic cocycles with non-trivial unstable bundles. [ABSTRACT FROM AUTHOR]

It is shown that the power set of κ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on 2κ under Borel reducibility. Here κ is an uncountable regular cardinal with κκ = κ. [ABSTRACT FROM AUTHOR]

Let Iccc be the s-ideal of subsets of the Cantor group 2N generated by Borel sets which belong to every translation-invariant s-ideal on 2N satisfying the countable chain condition (ccc). We prove that Iccc strongly violates ccc. This generalizes a theorem of Balcerzak-Roslanowski-Shelah stating the same for the s-ideal on 2Ngenerated by Borel sets B ⊆ 2N which have perfectly many pairwise disjoint translates. We show that the last condition does not follow from B?Iccc even if B is assumed to be compact. Various other conditions which for a Borel set B imply that B ? Iccc are also studied. As a consequence we prove in particular that: If An are Borel sets, n ? N, and 2N =⋃n An, then there is n0 such that every perfect set P ⊆ 2N has a perfect subset Q, a translate of which is contained in An0. CH is equivalent to the statement that 2N can be partitioned into ?1 many disjoint translates of a closed set. [ABSTRACT FROM AUTHOR]