Homogeneous Borel Sets

Homogeneous Borel Sets of Ambiguous Class Two

We describe and characterize all homogeneous subsets of the Cantor set which are both an F„s and a GSo; it turns out that there are wx such spaces.

A Course on Borel Sets

Borel Sets with Large Squares

For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ: if a Borel set B ⊆ R × R contains a μ-square (i.e. a set of the form A × A, |A| = μ) then it contains a 20 -square and even a perfect square. And also for ⊗′μ: if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute...

Generating Borel Sets by Balls

It is proved that an arbitrary infinite-dimensional Banach space with basis admits an equivalent norm such that any Borel set can be obtained from balls by taking complements and countable disjoint unions. For reflexive spaces, the new norm can be chosen arbitrarily close to the initial norm.

Borel Sets and Countable Models

We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alon...