Higher random indestructibility of MAD families

TL;DR

This paper characterizes when maximal almost disjoint families at a weakly compact cardinal are indestructible under higher random forcing, linking combinatorial properties to forcing indestructibility and set-theoretic invariants.

Contribution

It provides a combinatorial criterion for indestructibility of MAD families under higher random forcing at weakly compact cardinals, extending classical results.

Findings

Characterization of indestructibility using combinatorial properties

Equivalence of certain set-theoretic invariants in this context

Existence of indestructible MAD families under specific conditions

Abstract

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $\kappa$ is indestructible by the higher random forcing $\mathbb Q_\kappa$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_\kappa) = \mathfrak b_\kappa = \mathfrak c_\kappa$ implies the existence $\mathbb Q_\kappa$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.

Equations

$$\mathrm{add}({\mathrm{null}}_{\kappa })={\mathfrak{b}}_{\kappa }={\mathfrak{c}}_{\kappa }$$

$${\kappa }^{+}<\mathrm{add}({\mathrm{null}}_{\kappa })={\mathfrak{b}}_{\kappa }={\mathfrak{c}}_{\kappa }$$

$$[A]=\{\eta \in 2^{\kappa }:\text{there exist cofinally many }i<\kappa \text{ such that }\eta \upharpoonright i\in A\}\subseteq 2^{\kappa }.$$

$$2^{\kappa }\setminus A\supseteq \bigcap _{\mathcal{J}\in \Lambda }\bigcup _{p\in \mathcal{J}}[p].$$

$$B\in {\mathrm{null}}_{\kappa }\iff {⊩}_{{\mathbb{Q}}_{\kappa }}\dot{\eta }\notin B.\text{qed}$$

$$2^{\kappa }\setminus A\supseteq \bigcup _{p\in \mathcal{J}}[p].$$

$$\bigcup _{r\in \mathcal{J}}[r]\cap ([q]\setminus B)=\varnothing .$$

$$A=\{f(\eta \upharpoonright i):i<\kappa ,\eta \upharpoonright i\in B\}.$$

$$p⊩\text{“}\dot{x}\in [\kappa {]}^{\kappa }\text{” and }(\forall I\in \mathcal{I})p⊩\text{“}|\dot{x}\cap I|<\kappa \text{”}.$$

$$f(\rho )={\alpha }_{\rho }\text{ such that }{p}^{[\rho ]}⊩\text{“}{\dot{x}}_i={\alpha }_{\rho }\text{”}.$$

$$X=\bigcup _{ϵ<\zeta }[f_{\zeta }^{-1}[A_{ϵ}]]\in {\mathrm{null}}_{\kappa }.$$

$$X\cap \bigcup _{p\in \mathcal{J}}[p]=\varnothing .$$

$$|f_{\zeta }^{-1}[A_{{ϵ}_i}]\cap q^{[{\rho }_i]}|=\kappa .$$

$$[\bigcup _{j<i}{f}_{\zeta }^{-1}[A_{{ϵ}_j}]\cap q^{[{\rho }_i]}]=\varnothing$$

$$[q^{[{\rho }_i]}\setminus \bigcup _{j<i}{f}_{\zeta }^{-1}[A_{{ϵ}_j}]]=[q^{[{\rho }_i]}]$$

$$A_{\zeta }=f[q^{[{\rho }_i]}\setminus \bigcup _{j<i}{f}_{\zeta }^{-1}[A_{{ϵ}_j}]]$$

$$g_{ϵ}(i)=\sup (A_{ϵ}\cap A_{{ϵ}_i}).$$

$$k_i\in \{m\in A_{{ϵ}_i}:f_{\zeta }^{-1}[\{m\}]\cap q\cap \{\varrho :{\rho }_i⊴\varrho \}\ne \varnothing \wedge m>g(i)\}\setminus \bigcup _{j<i}{A}_{{ϵ}_j}.$$

Full text

1 Introduction

In this paper $\kappa$ refers to a weakly compact cardinal. A family $\mathcal{A}\subseteq[\kappa]^{\kappa}$ is called almost disjoint if for all distinct $A,B\in\mathcal{A}$ we have $|A\cap B|<\kappa$. An almost disjoint family $\mathcal{A}$ is called maximal if for no almost disjoint family $\mathcal{B}\subseteq[\kappa]^{\kappa}$ we have $\mathcal{A}\subsetneq\mathcal{B}$.

The following way of constructing a maximal almost disjoint family $\mathcal{A}^{*}$ of $\kappa$ suggests itself. Identify $\kappa$ with $2^{<\kappa}$ and for $\eta\in 2^{\kappa}$ let $A_{\eta}=\{\eta{\upharpoonright}i:i<\kappa\}\subseteq 2^{<\kappa}$. Using the Teichmüller-Tukey lemma we can extend $\{A_{\eta}:\eta\in 2^{\kappa}\}$ to a maximal almost disjoint family $\mathcal{A}^{*}$.

Let $\mathbb{P}$ be a forcing notion. We say that a maximal almost disjoint family $\mathcal{A}$ is $\mathbb{P}$-indestructible if $\mathcal{A}$ remains maximal in any $\mathbb{P}$-generic extension. It is easy to see that any forcing notion $\mathbb{P}$ adding a real $\eta\in 2^{\kappa}$ destroys the family $\mathcal{A}^{*}$ from above, if $\mathbb{P}$ satisfies Mostowski’s absoluteness.111 A forcing notion $\mathbb{P}$ satisfies Mostowski’s absoluteness if $\Sigma_{1}^{1}$ formulas are absolute between $V$ and $V^{\mathbb{P}}$. Any $\kappa$-strategically closed forcing has this property, see e.g. [\citeauthoryearFriedman, Khomskii, and KulikovFriedman et al.2016, Lemma 2.7] or [\citeauthoryearBaumhauerBaumhauer2019, Lemma 4.2.1]

This leads to the question: Given a forcing notion $\mathbb{P}$, does there exist a maximal almost disjoint family $\mathcal{A}$ such that $\mathcal{A}$ is $\mathbb{P}$-indestructible? For the classical case $\kappa=\omega$ [\citeauthoryearKunenKunen1980] shows that assuming CH there exists a Cohen-indestructible maximal almost disjoint family. [\citeauthoryearHrušákHrušák2001] and [\citeauthoryearKurilićKurilić2001] provide a combinatorial characterization of Cohen-indestructibility and [\citeauthoryearHrušákHrušák2001] also investigates Sacks and Miller forcing. [\citeauthoryearBrendle and YatabeBrendle and Yatabe2005] continue this line of research, investigating several classical forcing notions and in particular provide a combinatorial characterization of indestructibility for the classical random forcing.

We shall deal with $\mathbb{Q}_{\kappa}$-indestructibility, where $\mathbb{Q}_{\kappa}$ is the higher random forcing from [\citeauthoryearShelahShelah2017]. In Theorem 4.1 we give a combinatorial characterization of $\mathbb{Q}_{\kappa}$-indestructibility, parallel to the one in [\citeauthoryearBrendle and YatabeBrendle and Yatabe2005, Theorem 2.4.9.] for the classical random forcing. In Theorem 4.2 we use this characterization to show that

[TABLE]

implies the existence of a $\mathbb{Q}_{\kappa}$-indestructible maximal almost disjoint family of $\kappa$. Here $\operatorname{\mathbf{null}}_{\kappa}$ denotes the higher null ideal from [\citeauthoryearShelahShelah2017] (there referred to as $\operatorname{id}(\mathbb{Q}_{\kappa}))$ and $\mathfrak{c}_{\kappa}$ denotes the size of $2^{\kappa}$. This result is again parallel to [\citeauthoryearBrendle and YatabeBrendle and Yatabe2005, Theorem 3.6.1.] where it is shown that $\operatorname{add}(\operatorname{\mathbf{null}})=\mathfrak{c}$ implies the existence of a random indestructible maximal almost disjoint family.

Clearly $\mathfrak{c}_{\kappa}=\kappa^{+}$ implies ($*$ ‣ 1). However this assumption is not necessary, as the Amoeba model in [\citeauthoryearBaumhauer, Goldstern, and ShelahBaumhauer et al.2018, Section 6] shows (assuming $\kappa$ supercompact) that

[TABLE]

is consistent. Compared to the classical case we need the additional assumption $\mathfrak{b}_{\kappa}=\mathfrak{c}_{\kappa}$, as the consistency of $\operatorname{add}(\operatorname{\mathbf{null}}_{\kappa})>\mathfrak{b}_{\kappa}$ is an open problem.

2 Notation and Conventions

We use the following conventions. If $f:X\to Y$ is a function, $A\subseteq X$ and $B\subseteq Y$, then $f[A]=\{f(x):x\in A\}$ and $f^{-1}[B]=\{x\in X:f(x)\in B\}$. For $\rho\in 2^{<\kappa}$ let $[\rho]=\{\eta\in 2^{\kappa}:\rho\trianglelefteq\eta\}.$ For $\rho,\varrho\in 2^{<\kappa}$ we write $\rho\trianglelefteq\varrho$ if $\rho\subseteq\varrho$. For $\rho\in p\in\mathbb{Q}_{\kappa}$ let $p^{[\rho]}=\{\varrho\in p:\varrho\trianglelefteq\rho\lor\rho\trianglelefteq\varrho\}\in\mathbb{Q}_{\kappa}$.

For $A\subseteq 2^{<\kappa}$ define the $G_{\delta}$-closure

[TABLE]

(Note that if $A$ is downward closed (i.e. a tree), $[A]$ is a closed set.)

On $2^{\kappa}$ we use the topology generated by the basic clopen sets $[\rho]$ for $\rho\in 2^{<\kappa}$. The $\kappa$-Borel sets $\operatorname{\mathbf{Borel}}_{\kappa}$ are the smallest family containing all basic clopen sets which is closed under complements and unions/intersections of at most $\kappa$-many sets.

3 Higher Random Forcing

The higher random forcing $\mathbb{Q}_{\kappa}$ for a (strongly) inaccessible cardinal $\kappa$ was introduced by Saharon Shelah in [\citeauthoryearShelahShelah2017]. Recall that $\mathbb{Q}_{\kappa}$ is a tree forcing on $2^{<\kappa}$ with the following properties:

1. (a)

$\mathbb{Q}_{\kappa}$ satisfies the $\kappa^{+}$-chain condition. 2. (b)

$\mathbb{Q}_{\kappa}$ is strategically $\kappa$-closed. 3. (c)

If $\kappa$ is weakly compact, then $\mathbb{Q}_{\kappa}$ is $\kappa^{\kappa}$-bounding.

The higher null ideal $\operatorname{\mathbf{null}}_{\kappa}$ consists of all sets $A\subseteq 2^{\kappa}$ such that there exists a family $\Lambda$ of $\kappa$-many maximal antichains of $\mathbb{Q}_{\kappa}$ such that

[TABLE]

If $G$ is a $\mathbb{Q}_{\kappa}$-generic filter then we call $\eta=\bigcup_{p\in G}\operatorname{tr}(p)$ the $\mathbb{Q}_{\kappa}$-generic real or random real, where $\operatorname{tr}(p)$ is the trunk of $p$. Throughout the paper $\dot{\eta}$ will denote a name for the canonical generic real added by $\mathbb{Q}_{\kappa}$.

Fact 3.1**.**

Let $p,q\in\mathbb{Q}_{\kappa}$. The following are equivalent:

1. (i)

$p,q$ are compatible. 2. (ii)

$[p]\cap[q]\neq\emptyset$. 3. (iii)

$\operatorname{tr}(p)\trianglelefteq\operatorname{tr}(q)\in p\ \lor\ \operatorname{tr}(q)\trianglelefteq\operatorname{tr}(p)\in q$. ∎

Lemma 3.2**.**

Let $p,q\in\mathbb{Q}_{\kappa}$. Then:

1. (i)

If $q\Vdash\dot{\eta}\in[p]$, then $q\not\perp p.$ 2. (ii)

If $[q]\subseteq[p]$, then $q\leq p$

Proof.

1. (i)

Let $q\Vdash$ “$\dot{\eta}\in[p]$” and towards contradiction assume $q\perp p$. According to Fact 3.1 (iii) there are three cases:

1. (1)

$\operatorname{tr}(p)\perp\operatorname{tr}(q)$ 2. (2)

$\operatorname{tr}(p)\trianglelefteq\operatorname{tr}(q)\not\in p$ 3. (3)

$\operatorname{tr}(q)\trianglelefteq\operatorname{tr}(p)\not\in q$.

As an example consider case (2). For every $\nu\in[p]$ we have $\operatorname{tr}(q)\not\trianglelefteq\nu$. But clearly $q\Vdash$“$\operatorname{tr}(q)\trianglelefteq\dot{\eta}$”. Contradiction to $q\Vdash$ “$\dot{\eta}\in[p]$”.

Work similarly for case (1) and (3). 2. (ii)

Similarly.∎

Fact 3.3**.**

Let $B\in\operatorname{\mathbf{Borel}}_{\kappa}$. Then:

[TABLE]

This is shown in [\citeauthoryearShelahShelah2017, Claim 3.2] by induction on the Borel rank of $B$.

Fact 3.4**.**

For any $p\in\mathbb{Q}_{\kappa}$ we have $[p]\not\in\operatorname{\mathbf{null}}$. This is a simple consequence of the observation that $p\Vdash$“$\dot{\eta}\in[p]$” and Fact 3.3.∎

Fact 3.5**.**

Let $\kappa$ be weakly compact. If $A\in\operatorname{\mathbf{null}}_{\kappa}$, then there exists a single maximal antichain $\mathcal{J}$ of $\mathbb{Q}_{\kappa}$ such that

[TABLE]

This is shown in [\citeauthoryearBaumhauer, Goldstern, and ShelahBaumhauer et al.2018, Lemma 1.3.3., Lemma 3.1.2]. ∎

Theorem 3.6**.**

Let $\kappa$ be weakly compact. Let $B\in\operatorname{\mathbf{Borel}}_{\kappa}{\setminus}\operatorname{\mathbf{null}}_{\kappa}$. Then there exists $p\in\mathbb{Q}_{\kappa}$ such that $[p]\subseteq B$.

In words: every positive Borel set contains a random condition.

Proof.

By Fact 3.3 there exists $q$ such that $q\Vdash$“$\dot{\eta}\in B$”. Consider $[q]{\setminus}B$. There are two cases:

1. (1)

$[q]{\setminus}B\in\operatorname{\mathbf{null}}_{\kappa}$. By Fact 3.5 there exists a single maximal antichain $\mathcal{J}\subseteq\mathbb{Q}_{\kappa}$ such that

[TABLE]

Choose $r\in\mathcal{J}$ compatible with $q$. Then $p=r\land q$ is as required. 2. (2)

$[q]{\setminus}B\not\in\operatorname{\mathbf{null}}_{\kappa}$. By Fact 3.3 there exists $r\in\mathbb{Q}_{\kappa}$ such that $r\Vdash\dot{\eta}\in[q]{\setminus}B.$ So in particular

1. (a)

$r\Vdash\dot{\eta}\in[q]$. 2. (b)

$r\Vdash\dot{\eta}\not\in B$.

By (a) and Lemma 3.2(i) we have $r\not\perp q$. But by our choice of $q$ we have $q\Vdash$“$\dot{\eta}\in B$”, hence by (b) $q$ and $r$ cannot be compatible. Contradiction, i.e. this case does not appear.∎

4 Results

Any maximal almost disjoint family $\mathcal{A}$ canonically defines the ideal $\mathcal{I}(\mathcal{A})$ of all subsets of $\kappa$ that can be covered by ${<}\kappa$-many elements of $\mathcal{A}$. Let $\mathbb{P}$ be a forcing notion. We say $\mathcal{I}(\mathcal{A})$ is $\mathbb{P}$-indestructible if $\mathbb{P}$ does not add a pseudo-intersection to the dual filter of $\mathcal{I}(\mathcal{A})$. Easily $\mathcal{A}$ is $\mathbb{P}$-indestructible iff $\mathcal{I}(\mathcal{A})$ is $\mathbb{P}$-indestructible.

Theorem 4.1**.**

Let $\kappa$ be a weakly compact cardinal. Let $\mathcal{A}\subseteq[\kappa]^{\kappa}$ be a maximal almost disjoint family and let $\mathcal{I}=\mathcal{I}(\mathcal{A})$. The following are equivalent:

1. (i)

$\mathcal{I}$* is $\mathbb{Q}_{\kappa}$-indestructible.* 2. (ii)

$(\forall B\subseteq 2^{<\kappa},[B]\not\in\operatorname{\mathbf{null}}_{\kappa})(\forall f:B\rightarrow\kappa)(\exists I\in\mathcal{I})\ [f^{-1}[I]]\not\in\operatorname{\mathbf{null}}_{\kappa}$. 3. (iii)

$(\forall B\subseteq 2^{<\kappa},[B]\not\in\operatorname{\mathbf{null}}_{\kappa})(\forall f:B\rightarrow\kappa,f\text{ is }{<}\kappa\text{-to-one})(\exists I\in\mathcal{I})$

$[f^{-1}[I]]\not\in\operatorname{\mathbf{null}}_{\kappa}$.

Proof.

(i) $\Rightarrow$ (ii): Assume (ii) fails, i.e. there exist $B\subseteq 2^{<\kappa}$, $[B]\not\in\operatorname{\mathbf{null}}_{\kappa}$ and $f:B\to\kappa$ such that $[f^{-1}[I]]\in\operatorname{\mathbf{null}}_{\kappa}$ for all $I\in\mathcal{I}$. By Theorem 3.6 there exists $p\in\mathbb{Q}_{\kappa}$ such that $[p]\subseteq[B]$. Let $G$ be a $\mathbb{Q}_{\kappa}$-generic filter containing $p$ and let $\eta=\bigcup_{q\in G}\operatorname{tr}(q)\in 2^{\kappa}$, hence by Fact 3.3 we have $\eta\not\in[f^{-1}[I]]$ for all $I\in\mathcal{I}$. Consider

[TABLE]

First note that because $\eta\in[f^{-1}[A]]$ we have $A\not\in\mathcal{I}$. Without loss of generality $\bigcup\mathcal{A}=\kappa$, hence all sets of size less than $\kappa$ are contained in $\mathcal{I}$, which implies $|A|=\kappa$.

Now check that $A$ destroys $\mathcal{I}$. Assume it does not, i.e. there exists $I\in\mathcal{I}$ such that $|I\cap A|=\kappa$. This implies $\eta{\upharpoonright}i\in f^{-1}[I]$ for cofinally many $i<\kappa$, hence $\eta\in[f^{-1}[I]]$. Contradiction, thus $A$ is almost disjoint from all $I\in\mathcal{I}$, i.e. $p\Vdash$“$\mathcal{I}$ is destroyed”. So we have shown that $\lnot$(ii) implies $\lnot$(i).

(ii) $\Rightarrow$ (iii): Trivial.

(iii) $\Rightarrow$ (i): Towards contradiction assume there is $p\in\mathbb{Q}_{\kappa}$ and a $\mathbb{Q}_{\kappa}$-name $\dot{x}$ such that

[TABLE]

Furthermore let $p$ be a fusion condition as in [\citeauthoryearShelahShelah2017, Claim 1.9.], i.e. such that there exists a cofinal sequence $\langle\beta_{i}:i<\kappa\rangle$ such that for all $i<\kappa$, $\rho\in 2^{\beta_{i}}\cap p$ the condition $p^{[\rho]}$ decides $\dot{x}_{i}$, where $\langle\dot{x}_{i}:i<\kappa\rangle$ is an increasing enumeration of $\dot{x}$.

Let $B=p\cap\bigcup_{i<\kappa}2^{\beta_{i}}$ and clearly $[B]=[p]$, hence $B\not\in\operatorname{\mathbf{null}}_{\kappa}$ by Fact 3.4. Define $f:B\to\kappa$ such that for $\rho\in B\cap 2^{\beta_{i}}$ we have

[TABLE]

Is is easy to see that $f$ is ${<}\kappa$-to-one since our choice of $p$ implies $\rho\in B\cap 2^{\beta_{i}}\Rightarrow f(\rho)\geq i.$

By our assumption there exists $I\in\mathcal{I}$ such that $[f^{-1}[I]]\not\in\operatorname{\mathbf{null}}_{\kappa}$, hence by Theorem 3.6 there exists $q\in\mathbb{Q}_{\kappa}$ such that $[q]\subseteq[f^{-1}[I]]$. Of course $[f^{-1}[I]]\subseteq[p]$, hence $[q]\subseteq[p]$, and by Lemma 3.2(ii) this implies $q\leq p$.

But $q\Vdash$“$|\dot{x}\cap I|=\kappa$”. Contradiction. ∎

Note that the proof of Theorem 4.1 essentially verifies that $\mathbb{Q}_{\kappa}$ satisfies a $\kappa$-version of weak fusion as defined in [\citeauthoryearBrendle and YatabeBrendle and Yatabe2005, Definition 2.2.1] (except there a one-to-one function is required). However, as the definition of weak fusion is fairly technical, doing the proof directly may be more transparent.

Theorem 4.2**.**

Let $\kappa$ be a weakly compact cardinal. If $\operatorname{add}(\operatorname{\mathbf{null}}_{\kappa})=\mathfrak{b}_{\kappa}=\mathfrak{c}_{\kappa}$, then there exists a $\mathbb{Q}_{\kappa}$-indestructible maximal almost disjoint family $\mathcal{A}\subseteq[\kappa]^{\kappa}$ of size $\mathfrak{c}_{\kappa}$.

Proof.

Let $\langle(B_{\zeta},f_{\zeta}):\kappa\leq\zeta<\mathfrak{c}_{\kappa}\rangle$ enumerate all pairs $(B,f)$ where $B\subseteq 2^{<\kappa}$, $[B]\not\in\operatorname{\mathbf{null}}_{\kappa}$ and $f:B\to\kappa$ is a ${<}\kappa$-to-one function. Let $\langle A_{\zeta}:\zeta<\kappa\rangle$ be a partition of $\kappa$ into sets of size $\kappa$. We are inductively going to construct sequence $\langle A_{\zeta}:\kappa\leq\zeta<\mathfrak{c}_{\kappa}\rangle$ such that for all $\zeta\in[\kappa,\mathfrak{c}_{\kappa})$:

1. (1)

$A_{\zeta}\in[\kappa]^{\kappa}$. 2. (2)

$(\forall\epsilon<\zeta)\ |A_{\zeta}\cap A_{\epsilon}|<\kappa$. 3. (3)

$(\exists\epsilon\leq\zeta)\ [f_{\zeta}^{-1}[A_{\epsilon}]]\not\in\operatorname{\mathbf{null}}_{\kappa}$.

If we can carry out this construction, we may find a maximal almost disjoint family $\mathcal{A}\supseteq\{A_{\zeta}:\zeta<\mathfrak{c}_{\kappa}\}$ using the Teichmüller-Tukey lemma, and $\mathcal{A}$ is $\mathbb{Q}_{\kappa}$-indestructible by Theorem 4.1.

At stage $\zeta$ consider $f_{\zeta}:p_{\zeta}\to\kappa$.

Case 1: There exists $\epsilon<\zeta$ such that $[f_{\zeta}^{-1}[A_{\epsilon}]]\not\in\operatorname{\mathbf{null}}_{\kappa}$. In this case let $A_{\zeta}$ be any set satisfying (1) and (2). Remember $\zeta<\mathfrak{c}_{\kappa}=\mathfrak{b}_{\kappa}\leq\mathfrak{a}_{\kappa}$ so this is always possible.

Case 2: For all $\epsilon<\zeta$ we have $[f_{\zeta}^{-1}[A_{\epsilon}]]\in\operatorname{\mathbf{null}}_{\kappa}$. By Theorem 3.6 there exists $p_{\zeta}\in\mathbb{Q}_{\kappa}$ such that $[p_{\zeta}]\subseteq[B_{\zeta}]$. By our assumption $\zeta<\operatorname{add}(\operatorname{\mathbf{null}}_{\kappa})$, hence also

[TABLE]

By Fact 3.5 there exists a maximal antichain $\mathcal{J}$ of $\mathbb{Q}_{\kappa}$ such that

[TABLE]

Let $p\in\mathcal{J}$ be such that $p\not\perp p_{\zeta}$ and let $q=p\cap p_{\zeta}$. Clearly $X\cap[q]=\emptyset$.

Now the plan is as follows: $f_{\zeta}[q\cap B_{\zeta}]$ is a candidate for $A_{\zeta}$ satisfying (1) and (3). So we want to thin out $f_{\zeta}[q\cap B_{\zeta}]$ to some $A_{\zeta}\subseteq f_{\zeta}[q\cap B]$ satisfying (2) and still satisfying (1) and (3). We use a combinatorial argument from [\citeauthoryearHrušákHrušák2001] to finish the proof.

Let $\langle\rho_{i}:i<\kappa\rangle$ enumerate $q\cap B_{\zeta}$. For $i<\kappa$ inductively try to choose distinct $\epsilon_{i}<\zeta$ such that

[TABLE]

If this construction fails at stage $i<\kappa$ note that

[TABLE]

hence

[TABLE]

and easily

[TABLE]

is as required, i.e. $A_{\zeta}$ is almost disjoint from $A_{\epsilon}$ for all $\epsilon<\zeta$ and $[f^{-1}_{\zeta}[A_{\zeta}]]\supseteq[q^{[\rho_{i}]}]$, hence $[f^{-1}_{\zeta}[A_{\zeta}]]\not\in\operatorname{\mathbf{null}}_{\kappa}$ by Fact 3.4.

So assume the construction succeeded and for $\epsilon\in\zeta{\setminus}\{\epsilon_{i}:i<\kappa\}$ define $g_{\epsilon}:\kappa\to\kappa$ by

[TABLE]

Remember $\zeta<\mathfrak{c}_{\kappa}=\mathfrak{b}_{\kappa}$ and find $g\in\kappa^{\kappa}$ such that $g_{\epsilon}\leq^{*}g$ for all $\epsilon$. Now for every $i<\kappa$ choose

[TABLE]

Let $A_{\zeta}=\{k_{i}:i<\kappa\}$. By construction $A_{\zeta}$ is almost disjoint from $A_{\epsilon}$ for all $\epsilon<\zeta$ and $[f_{\zeta}^{-1}[A_{\zeta}]]\supseteq[q]$, hence $[f_{\zeta}^{-1}[A_{\zeta}]]\not\in\operatorname{\mathbf{null}}_{\kappa}$ by Fact 3.4. ∎

Acknowledgements

I thank Martin Goldstern, who read this manuscript and provided valuable comments, suggestions and corrections.

I thank Vera Fischer, who during the defense of my thesis asked the question of the existence of a higher random indestructible maximal almost disjoint family, and who thus inspired this paper.