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.

中文翻译：

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受限制的疯狂家庭

让${\卡尔我}$成为一个理想的人ω. 由 cov${}_{}^{\rm{*}}({\cal I})$我们表示一个家庭的最小规模${\cal B} \subseteq {\cal I}$这样对于每一个无限$X \in {\cal I}$有$B \in {\cal B}$为此$B\mathop \cap \nolimits X$是无限的。我们说一个AD家族${\cal A} \subseteq {\cal I}$是一个MAD 家族仅限于${\卡尔我}$如果对于每个无限$X \in {\cal I}$有$A \in {\cal A}$这样$|X\mathop \cap \nolimits A| = \欧米茄$. 让一个$\left( {\cal I} \right)$是无限 MAD 家族的最小规模，限制为${\卡尔我}$. 我们证明如果$最大${a,cov${}_{}^{\rm{*}}({\cal I})\}$然后一个$\left( {\cal I} \right) = {\omega _1}$，因此，如果${\卡尔我}$很高，而且$\le {\omega _2}$然后一个$\left( {\cal I} \right) = max${a,cov${}_{}^{\rm{*}}({\cal I})\}$. 我们用这些结果来证明如果 c$\le {\omega _2}$然后$= \overline o$那一个s$= 最大值${一个，非$({\cal M})\}$. 我们还分析了每个AD家族的大小是否与CH的否定一致的问题ω1可以扩展到 MAD 系列的大小ω1.