RESTRICTED MAD FAMILIES

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.