RESTRICTED MAD FAMILIES

• Published in: The Journal of symbolic logic 2020-03, Vol.85 (1), p.149-165
• Main Authors: GUZMÁN, OSVALDO, HRUŠÁK, MICHAEL, TÉLLEZ, OSVALDO
• Format: Article
• Language: English
• Subjects: Logic; Mathematics; Set theory
• ISSN: 0022-4812; 1943-5886
• DOI: 10.1017/jsl.2019.76

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 a s $= max$ {a,non $({\cal M})\}$ . We also analyze the problem whether it is consistent with the negation of CH that every AD family of size ω 1 can be extended to a MAD family of size ω 1 .