1 5 M ay 1 99 6 CARDINAL PRESERVING IDEALS

ar X iv : m at h / 97 10 21 7 v 1 [ m at h . L O ] 1 5 O ct 1 99 7 Ultrafilters on ω — — their ideals and their cardinal characteristics

For a free ultrafilter U on ω we study several cardinal characteristics which describe part of the combinatorial structure of U . We provide various consistency results; e.g. we show how to force simultaneously many characters and many π–characters. We also investigate two ideals on the Baire space ω naturally related to U and calculate cardinal coefficients of these ideals in terms of cardinal...

ar X iv : m at h / 04 05 08 1 v 1 [ m at h . L O ] 5 M ay 2 00 4 Preserving Preservation

We present preservation theorems for countable support iteration of nep forcing notions satisfying “old reals are not Lebesgue null” (section 6) and “old reals are not meager” (section 5). (Nep is a generalization of Suslin proper.) We also give some results for general Suslin ccc ideals (the results are summarized in a diagram on page 17). This paper is closely related to [She98, XVIII, §3] an...

2 0 M ay 1 99 8 EFFECTIVE NULLSTELLENSATZ FOR ARBITRARY IDEALS

ar X iv : m at h / 97 05 22 6 v 1 [ m at h . L O ] 1 5 M ay 1 99 7 On a problem of Steve Kalikow

The Kalikow problem for a pair (λ, κ) of cardinal numbers, λ > κ (in particular κ = 2) is whether we can map the family of ω–sequences from λ to the family of ω–sequences from κ in a very continuous manner. Namely, we demand that for η, ν ∈ ωλ we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer e.g. for אω but we prove it for smaller cardin...

ar X iv : m at h / 99 05 12 0 v 1 [ m at h . L O ] 1 9 M ay 1 99 9 BOREL IMAGES OF SETS OF REALS

The main goal of this paper is to generalize several results concerning cardinal invariants to the statements about the associated families of sets. We also discuss the relationship between the additive properties of sets and their Borel images. Finally, we present estimates for the size of the smallest set which is not strongly meager.