7 research outputs found
On weakly tight families
Using ideas from Shelah's recent proof that a completely separable maximal
almost disjoint family exists when , we construct a
weakly tight family under the hypothesis \s \leq \b < {\aleph}_{\omega}. The
case when \s < \b is handled in \ZFC and does not require \b <
{\aleph}_{\omega}, while an additional PCF type hypothesis, which holds when
\b < {\aleph}_{\omega} is used to treat the case \s = \b. The notion of a
weakly tight family is a natural weakening of the well studied notion of a
Cohen indestructible maximal almost disjoint family. It was introduced by
Hru{\v{s}}{\'a}k and Garc{\'{\i}}a Ferreira \cite{Hr1}, who applied it to the
Kat\'etov order on almost disjoint families
Universally Sacks-indestructible combinatorial families of reals
We introduce the notion of an arithmetical type of combinatorial family of
reals, which serves to generalize different types of families such as mad
families, maximal cofinitary groups, ultrafilter bases, splitting families and
other similar types of families commonly studied in combinatorial set theory.
We then prove that every combinatorial family of reals of arithmetical type,
which is indestructible by the product of Sacks forcing
, is in fact universally Sacks-indestructible, i.e. it
is indestructible by any countably supported iteration or product of
Sacks-forcing of any length. Further, under we present a unified
construction of universally Sacks-indestructible families for various
arithmetical types of families. In particular we prove the existence of a
universally Sacks-indestructible maximal cofinitary group under .Comment: 33 pages, submitte
Forcing indestructibility of MAD families
AbstractLet A⊆[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions (P,Q). We close with a detailed investigation of iterated Sacks indestructibility