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1. CJM 2012 (vol 64 pp. 1378)
On Weakly Tight Families Using ideas from Shelah's recent proof that a completely
separable maximal almost disjoint family exists when
$\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family
under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt
{\aleph}_{\omega}$.
The case when $\mathfrak{s} \lt \mathfrak{b}$
is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$,
while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt
{\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. 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 HruÅ¡Ã¡k and GarcÃa
Ferreira, who applied it to the KatÃ©tov order on almost
disjoint families.
Keywords:maximal almost disjoint family, cardinal invariants Categories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65 |
2. CJM 2011 (vol 63 pp. 1416)
MAD Saturated Families and SANE Player We throw some light on the question: is there a MAD family
(a maximal family of infinite subsets of $\mathbb{N}$, the intersection of any
two is finite) that is saturated (completely separable \emph{i.e.,} any
$X \subseteq \mathbb{N}$ is
included in a finite union of members of the family \emph{or} includes a
member (and even continuum many members) of the family).
We prove that it is hard to prove the consistency of the negation:
(i) if $2^{\aleph_0} \lt \aleph_\omega$, then there is such a family;
(ii) if there is no such family, then some situation
related to pcf holds whose consistency is large (and if ${\mathfrak a}_* \gt
\aleph_1$ even unknown);
(iii) if, \emph{e.g.,} there is no inner model with measurables,
\emph{then} there is such a family.
Keywords:set theory, MAD families, pcf, the continuum Categories:03E05, 03E04, 03E17 |