184 research outputs found
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,
Many Different Uniformity Numbers of Yorioka Ideals
Using a countable support product of creature forcing posets, we show that
consistently, for uncountably many different functions the associated Yorioka
ideals' uniformity numbers can be pairwise different. In addition we show that,
in the same forcing extension, for two other types of simple cardinal
characteristics parametrised by reals (localisation and anti-localisation
cardinals), for uncountably many parameters the corresponding cardinals are
pairwise different.Comment: 29 pages, 4 figure
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