Restricted Mad Families

Abstract

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.

Tight Eventually Different Families

“…As a result we obtain the consistency of a e = a p = d < a T . The proof of the preservation result mirrors the analogous one for tight MAD families given in [11,Proposition 38]. We recall some terminology used there.…”

“…The purpose of this section is to prove a preservation theorem for tight, eventually different families, akin to [11,Corollary 32] which showed the same for tight MAD families. The preservation theorem we prove concerns the notion of strong preservation.…”

“…The same can be said in the case of a p with the additional stipulation that every element is a bijection of ω with itself. For more examples of relatives of a see [11,13].…”

“…An almost disjoint family A is tight if given any countable set {B n | n < ω} of I(A) + sets there is a single C ∈ I(A) so that B n ∩ C is infinite for all n < ω. Tight MAD families exist under b = 2 ℵ 0 (see [10]), are Cohen indestructible ([10, Corollary 3.2]) and, under certain conditions are preserved by countable support iterations of proper forcing notions (see [11]).…”

“…To describe this model we recall the partition forcing introduced in [15] (see also [6] and [11,Section 1.6]). This forcing will be discussed in more length in Section 6.…”