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A Special Class of Almost Disjoint Families
The collection of branches (maximal linearly ordered sets of nodes) of the
tree (ordered by inclusion) forms an almost disjoint
family (of sets of nodes). This family is not maximal -- for example, any level
of the tree is almost disjoint from all of the branches. How many sets must be
added to the family of branches to make it maximal? This question leads to a
series of definitions and results: a set of nodes is {\it off-branch} if it is
almost disjoint from every branch in the tree; an {\it off-branch family} is an
almost disjoint family of off-branch sets; {\frak o}=\min\{|{\Cal O}|: {\Cal
O} is a maximal off-branch family. Results concerning include:
(in ZFC) , and (consistent with ZFC) is not
equal to any of the standard small cardinal invariants , ,
, or . Most of these consistency results use
standard forcing notions -- for example, comes from starting with a model of and
adding -many Cohen reals. Many interesting open questions remain,
though -- for example,
Maximal almost disjoint families, determinacy, and forcing
We study the notion of -MAD families where is a
Borel ideal on . We show that if is an arbitrary
ideal, or is any finite or countably iterated Fubini product of
ideals, then there are no analytic infinite -MAD
families, and assuming Projective Determinacy there are no infinite projective
-MAD families; and under the full Axiom of Determinacy +
there are no infinite -mad families.
These results apply in particular when is the ideal of finite sets
, which corresponds to the classical notion of MAD families. The
proofs combine ideas from invariant descriptive set theory and forcing.Comment: 40 page
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