On nonmeasurable unions

For a Polish space X and a σ-ideal I of subsets of X which has a Borel base we consider families A of sets in I with the union ?A not in I. We determine several conditions on A which imply the existence of a subfamily A^′ of A whose union ?A^′ is not in the σ-field generated by the Borel sets on X and I. Main examples are X=R and I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.     

Acyclicity and reduction

We provide, for each non-self dual Borel class Γ, a concrete finite antichain basis for the class of non-potentially Γ Borel relations whose closure has an acyclic symmetrization, considering the quasi-order of injective continuous reducibility. Along similar lines, we provide a sufficient condition for reducing the oriented graph ${\mathbb{G}}_0$ involved in the Kechris–Solecki–Todor?evi? dichotomy. We also prove a similar result giving a minimum set instead of an antichain if we allow rectangular reductions.     

A Fubini theorem

Let I₀ be the σ-ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I₀ and the σ-ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak-Ros?anowski-Shelah stating that I₀ on N² strongly violates the countable chain condition.     

Injective tests of low complexity in the plane

We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.     

On completely nonmeasurable unions

Assume that there is no quasi-measurable cardinal not greater than 2^ω . We show that for a c. c. c. σ -ideal 𝕀 with a Borel base of subsets of an uncountable Polish space, if 𝒜 is a point-finite family of subsets from 𝕀, then there is a subfamily of 𝒜 whose union is completely nonmeasurable, i.e. its intersection with every non-small Borel set does not belong to the σ -field generated by Borel sets and the ideal 𝕀. This result is a generalization of the Four Poles Theorem (see [1]) and a result from [3]. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Short separating geodesics for multiply connected domains

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest geodesic always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a meridian of the domain. We prove that, although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible geodesics which separate the complement in this fashion.     

Curves which Intersect Lines in Finite Sets

We study closed subsets in the plane which intersect each linein at least m points and at most n points, for which we tryto minimize the difference n – m. It is known that m cannotbe equal to n. The results in this paper show that for everyeven number n there exist closed sets in the plane for whichm = n – 2.     

Weak Borel chromatic numbers

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G ‐independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge. We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ?₁. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA. Slightly weaker results hold for F_σ‐graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable F_σ‐graph has a Borel chromatic number of at most ?₁. We refute various reasonable generalizations of these results to hypergraphs (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type

We derive ? ^r (μ)-bracketing metric and sup-norm metric entropy rates of bounded subsets of general function spaces defined over ? ^d or, more generally, over Borel subsets thereof, by adapting results of Haroske and Triebel (Math. Nachr. 167, 131–156, 1994; 278, 108–132, 2005). The function spaces covered are of (weighted) Besov, Sobolev, Hölder, and Triebel type. Applications to the theory of empirical processes are discussed. In particular, we show that (norm-)bounded subsets of the above mentioned spaces are Donsker classes uniformly in various sets of probability measures.     

Forcing with quotients

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.     

Continuous dependence on obstacles for the double obstacle problem on metric spaces

Let X be a complete metric space equipped with a doubling Borel measure supporting a p-Poincaré inequality. We obtain various convergence results for solutions of double obstacle problems on open subsets of X. In particular, we consider a sequence of double obstacle problems with converging obstacles and show that the corresponding solutions converge as well. We use the convergence properties to define and study a generalized solution of the double obstacle problem.     

On the Borelness of the intersection operation

We study the intersection operation of closed linear subspaces in a separable Banach space. We show that if the ambient space is quasi-reflexive, then the intersection operation is Borel. On the other hand, if the space contains a closed subspace with a Schauder decomposition into infinitely many non-reflexive spaces, then the intersection operation is not Borel. As a corollary, for a closed subspace of a Banach space with an unconditional basis, the intersection operation of the closed linear subspaces is Borel if and only if the space is reflexive. We also consider the intersection operation of additive subgroups in an infinite-dimensional separable Banach space, and show that if this intersection operation is Borel then the space is hereditarily indecomposable.     

Generating closed 2-cell embeddings in the torus and the projective plane

If a graphG is embedded in a manifoldM such that all faces are cells bounded by simple closed curves we say that this is a closed 2-cell embedding ofG inM. We show how to generate the 2-cell embeddings in the projective plane from two minimal graphs and the 2-cell embeddings in the torus from six minimal graphs by vertex splitting and face splitting.     

Countable connected Hausdorff and Urysohn bunches of arcs in the plane

In this paper, we answer a question by Krasinkiewicz, Reńska and Sobolewski by constructing countable connected Hausdorff and Urysohn spaces as quotient spaces of bunches of arcs in the plane. We also consider a generalization of graphs by allowing vertices to be continua and replacing edges by not necessarily connected sets. We require only that two “vertices” be in the same quasi-component of the “edge” that contains them. We observe that if a graph G cannot be embedded in the plane, then any generalized graph modeled on G is not embeddable in the plane. As a corollary we obtain not planar bunches of arcs with their natural quotients Hausdorff or Urysohn. This answers another question by Krasinkiewicz, Reńska and Sobolewski.     

Axial Borel functions

A function from the plane to the plane is axial if it does not change one coordinate. We show that every Borel permutation of the plane is a superposition of 11 Borel axial permutations.     

Maps of Borel sets

Questions concerning the structure of Borel sets were raised in special cases by Luzin, Aleksandrov, and Uryson as the problems of distinguishing the sets with certain homogeneous properties in Borel classes and determining the number of such pairwise nonhomemorphic sets. The universal homogeneity, i.e., the property to contain an everywhere closed copy of any Borel set of the same or smaller class, was considered by L.V. Keldysh. She called the sets of classes Π _α ⁰ , α > 2, of first category in themselves that possess this homogeneity property canonical and proved their uniqueness. Thus she revealed the central role of the universality property when describing homeomorphic Borel sets. These investigations led her to the problem of universality of Borel sets and to the problem of finding conditions under which there exists an open map between Borel sets. In this paper, such conditions are presented and similar questions are considered for closed, compact-covering, harmonious, and other stable maps.     

The chromatic polynomial of fatgraphs and its categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás-Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.     

Approximate decidability in euclidean spaces

We study concepts of decidability (recursivity) for subsets of Euclidean spaces ?^k within the framework of approximate computability (type two theory of effectivity). A new notion of approximate decidability is proposed and discussed in some detail. It is an effective variant of F. Hausdorff's concept of resolvable sets, and it modifies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of sets can equivalently be expressed by computability of the characteristic functions by means of appropriately working oracle Turing machines. The notion fulfills some natural requirements and is hereditary under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidability can be applied to investigate concepts of reducibility for subsets of Euclidean spaces.     

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.