For what infinite cardinals κ is there a partition of the real line R into precisely κ Borel sets? Hausdorff famously proved that there is a partition of R into א1 Borel sets. But other than this, we show that the spectrum of possible sizes of partitions of R into Borel sets can be fairly arbitrary. For example, given any A ⊆ ω with 0, 1 ∈ A, there is a forcing extension in which A = {n : there is a partition of R into אn Borel sets}. We also look at the corresponding question for partitions of… Expand

From some point of view Miller's modification of Sacks' forcing (from [2]) is the “minimal” one able to destroy a partition of ω ω into compact sets.Expand

The theorem generalizes a result of Brendle, Spinas and Zhang regarding the possible sizes of maximal cofinitary groups and establishes the consistency of , which for is due to Yi Zhang.Expand

The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω, and a complete solution to this problem is given under the assumption that vartheta = Â£1 and min = 1.Expand

We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2… Expand

New ♦-like principle ♦d consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that ¬♦d is consistent with CH and that in many models of d = ω1 the principle… Expand

The set of possible sizes of maximal independent families to which mif abbreviates maximal independent family is referred as spectrum of independence and Spec(mif) is designated.Expand