Generating Borel sets by balls

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.

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...

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...

Optimal core-sets for balls

Given a set of points P ⊂ R and value > 0, an core-set S ⊂ P has the property that the smallest ball containing S is within of the smallest ball containing P . This paper shows that any point set has an -core-set of size d1/ e, and this bound is tight in the worst case. A faster algorithm given here finds an core-set of size at most 2/ . These results imply the existence of small core-sets for ...