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RESTRICTED MAD FAMILIES
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 05 November 2019, pp. 149-165
- Print publication:
- March 2020
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- Article
- Export citation
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Let
${\cal I}$ be an ideal on ω. By cov
${}_{}^{\rm{*}}({\cal I})$ we denote the least size of a family 
${\cal B} \subseteq {\cal I}$ such that for every infinite 
$X \in {\cal I}$ there is 
$B \in {\cal B}$ for which 
$B\mathop \cap \nolimits X$ is infinite. We say that an AD family 
${\cal A} \subseteq {\cal I}$ is a MAD family restricted to
${\cal I}$ if for every infinite 
$X \in {\cal I}$ there is 
$A \in {\cal A}$ such that 
$|X\mathop \cap \nolimits A| = \omega$. Let a
$\left( {\cal I} \right)$ be the least size of an infinite MAD family restricted to 
${\cal I}$. We prove that If 
$max${a,cov
${}_{}^{\rm{*}}({\cal I})\}$ then a
$\left( {\cal I} \right) = {\omega _1}$, and consequently, if 
${\cal I}$ is tall and 
$\le {\omega _2}$ then a
$\left( {\cal I} \right) = max$ {a,cov
${}_{}^{\rm{*}}({\cal I})\}$. We use these results to prove that if c
$\le {\omega _2}$ then o
$= \overline o$ and that as
$= max${a,non
$({\cal M})\}$. We also analyze the problem whether it is consistent with the negation of CH that every AD family of size ω1 can be extended to a MAD family of size ω1.
