2 results
Splitting Families and Complete Separability
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 1 / 14 March 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 119-124
- Print publication:
- 14 March 2014
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We answer a question from Raghavan and Steprāns by showing that
$\mathfrak{s}\,\text{=}\,{{\mathfrak{s}}_{\omega ,\omega }}.$ Then we use this to construct a completely separable maximal almost disjoint family under 
$\mathfrak{s}\,\le \,\mathfrak{a}$, partially answering a question of Shelah.
On Weakly Tight Families
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- Journal:
- Canadian Journal of Mathematics / Volume 64 / Issue 6 / 01 December 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1378-1394
- Print publication:
- 01 December 2012
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Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when
$c<{{\aleph }_{\omega }}$, we construct a weakly tight family under the hypothesis 
$\mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$. The case when 
$\mathfrak{s}<\mathfrak{b}$ is handled in ZFC and does not require 
$\mathfrak{b}<{{\aleph }_{\omega }}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b}<{{\aleph }_{\omega }}$ 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 [8], who applied it to the Katétov order on almost disjoint families.
