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Search: MSC category 03E40 ( Other aspects of forcing and Boolean-valued models )

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1. CJM 2016 (vol 69 pp. 502)

Fischer, Vera; Mejia, Diego Alejandro
Splitting, Bounding, and Almost Disjointness Can Be Quite Different
We prove the consistency of $$ \operatorname{add}(\mathcal{N})\lt \operatorname{cov}(\mathcal{N}) \lt \mathfrak{p}=\mathfrak{s} =\mathfrak{g}\lt \operatorname{add}(\mathcal{M}) = \operatorname{cof}(\mathcal{M}) \lt \mathfrak{a} =\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c} $$ with $\mathrm{ZFC}$, where each of these cardinal invariants assume arbitrary uncountable regular values.

Keywords:cardinal characteristics of the continuum, splitting, bounding number, maximal almost-disjoint families, template forcing iterations, isomorphism-of-names
Categories:03E17, 03E35, 03E40

2. CJM 2012 (vol 64 pp. 1378)

Raghavan, Dilip; Steprāns, Juris
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 Hrušák and García Ferreira, who applied it to the Katétov order on almost disjoint families.

Keywords:maximal almost disjoint family, cardinal invariants
Categories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65

3. CJM 2007 (vol 59 pp. 575)

Hernández-Hernández, Fernando; Hrušák, Michael
Cardinal Invariants of Analytic $P$-Ideals
We study the cardinal invariants of analytic $P$-ideals, concentrating on the ideal $\mathcal{Z}$ of asymptotic density zero. Among other results we prove $ \min\{ \mathfrak{b},\cov\ (\mathcal{N}) \} \leq\cov^{\ast}(\mathcal{Z}) \leq\max\{ \mathfrak{b},\non(\mathcal{N}) \right\}. $

Categories:03E17, 03E40

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