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We prove that ω1 → (ω + ω + 1, n)3 for all n < ω. [ABSTRACT FROM AUTHOR]

In this paper we construct a complete injective holomorphic immersion C → C² whose image is dense in C². The analogous result is obtained for any closed complex submanifold X ⊂ Cn for n > 1 in place of C ⊂ C². We also show that if X intersects the unit ball Bn of Cn and K is a connected compact subset of X ∩ Bn, then there is a Runge domain Ω ⊂ X containing K which admits a complete injective holomorphic immersion Ω → Bn whose image is dense in Bn. [ABSTRACT FROM AUTHOR]

Let Γ be a discrete group. Following Linnell and Schick one can define a continuous ring c(Γ) associated with Γ. They proved that if the Atiyah Conjecture holds for a torsion-free group Γ, then c(Γ) is a skew field. Also, if Γ has torsion and the Strong Atiyah Conjecture holds for Γ, then c(Γ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = ℤ2≀ℤ. It is known that ℂ(ℤ2≀ℤ) does not even have a classical ring of quotients. Our main result is that if H is amenable, then c(ℤ2≀ H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930s. [ABSTRACT FROM AUTHOR]

We determine lower and exact estimates of Kolmogorov, Gelfand and linear n-widths of unit balls in Sobolev norms in Lp-spaces on compact Riemannian manifolds. As it was shown by us previously these lower estimates are exact asymptotically in the case of compact homogeneous manifolds. The proofs rely on two-sides estimates for the near-diagonal localization of kernels of functions of elliptic operators. [ABSTRACT FROM AUTHOR]

We prove that, assuming large cardinals, it is consistent that there are many singular cardinals μ, such that every stationary subset μ+ of reflects but there are stationary subsets of μ+ that do not reflect at ordinals of arbitrarily high cofmality. This answers a question raised by Todd Eisworth. [ABSTRACT FROM AUTHOR]

We show that Friedman's finite adjacent Ramsey theorem is unprovable in Peano Arithmetic, and give a new proof of the unprovability of the Paris--Harrington theorem in PA. We also determine the status of these theorems for each dimension. It is to be noted that the finite adjacent Ramsey theorem for dimension d is equivalent to the Paris--Harrington theorem for dimension d + 1 . [ABSTRACT FROM AUTHOR]

We examine the dimensions of the intersection of a subset E of an m-ary Cantor space Cm with the image of a subset F under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically max{dim E+dim F--dim Cm, 0}, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument. [ABSTRACT FROM AUTHOR]

We introduce a geometric generalization of Hall's marriage theorem. For any family F = {X1,..., Xm} of finite sets in Rd, we give conditions under which it is possible to choose a point Xi £ Xi for every 1 ≤ i ≤ m in such a way that the points {x¹, . . ., xm} C Rd are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell's celebrated generalization of Hall's theorem. [ABSTRACT FROM AUTHOR]

We show that Friedman's finite adjacent Ramsey theorem is unprovable in Peano Arithmetic, and give a new proof of the unprovability of the Paris--Harrington theorem in PA. We also determine the status of these theorems for each dimension. It is to be noted that the finite adjacent Ramsey theorem for dimension d is equivalent to the Paris--Harrington theorem for dimension d + 1 . [ABSTRACT FROM AUTHOR]

We prove that every sectional-Anosov flow of a compact 3-manifold M exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of M. Applications include a characterization of essential hyperbolicity, sensitivity to the initial conditions and a relationship between the topology of M and the denseness of the basin of the singularities. [ABSTRACT FROM AUTHOR]

Given a set E in ℂm and a point p ∈ E, there is a unique smallest complex-analytic germ Xp containing Ep, called the holomorphic closure of Ep. We study the holomorphic closure of semialgebraic arc-symmetric sets. Our main application concerns CR-continuation of semialgebraic arc-analytic mappings: A mapping f : M → ℂn on a connected real-analytic CR manifold which is semialgebraic arc-analytic and CR on a non-empty open subset of M is CR on the whole M. [ABSTRACT FROM AUTHOR]

Let S be a closed Riemann surface of genus g(≧2) and set Ṡ = S\{ẑ0} . Then we have the composed map φ o r of a map r : T (S) x U→ F(S) and the Bers isomorphism φ : F(S) → T (Ṡ), where F(S) is the Bers fiber space of S, T(X) is the Teichmüller space of X and U is the upper half-plane. The purpose of this paper is to show that the map φ o r : T(S) x U → T(Ṡ) has a continuous extension to some subset of the boundary T(S) x δU. [ABSTRACT FROM AUTHOR]

We introduce a notion of strong algebrability of subsets of linear algebras. Our main results are the following. The set of all sequences from c0 which are not summable with any power is densely strongly c-algebrable. The set of all sequences in l∞ whose sets of limit points are homeomorphic to the Cantor set is comeager and strongly c-algebrable. The set of all nonmeasurable functions from RR is 2c-algebrable. These results complete several by other authors, within the modern context of lineability. [ABSTRACT FROM AUTHOR]

A {\em coloring} of a set $X$ is any subset $C$ of $[X]^N$, where $N>1$ is a natural number. We give some sufficient conditions for the existence of a perfect $C$-homogeneous set, in the case where $C$ is $G_\delta$ and $X$ is a Polish space. In particular, we show that it is sufficient that there exist $C$-homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect $3$-clique if it contains any uncountable $k$-clique, where $k$ is a natural number or $\aleph_0$ (a set $K$ is a {\em $k$-clique} in $X$ if the convex hull of any of its $k$-element subsets is not contained in $X$). [ABSTRACT FROM AUTHOR]

A correction to the article "A Gysin formula for Hall-Littlewood polynomials" that was published in the 2015 issue.