Let $\mathcal{A}$ be an uncountable almost disjoint family (not necessarily maximal) of infinite subsets of $\mathbb{N}$. Denote by $\mathcal{A}\_{\subseteq}=\{ B\subseteq\mathbb{N}:|B|=\omega \wedge \exists A\in\mathcal{A}(B\subseteq A) \}$.


Question 1: Must there exist a $B\in\mathcal{A}\_{\subseteq}$ such that for any $n$, $\{C\in\mathcal{A}:|C\cap B|\geq n\}$ is uncountable?


If this is true, then:


Question 1': Must there exist a $B\in\mathcal{A}\_{\subseteq}$ such that for any $n$, $\{C\in\mathcal{A}:B\upharpoonright n\subset C\}$ is uncountable? (By $B\upharpoonright n$, I mean the first $n$ elements of $B$ listed in increasing order.)


I don't have an intuition as to whether this should be true or false, but the question stems from the observation that for any uncountable family $\mathcal{A}$ of subsets of $\mathbb{N}$, there is an $n\in\mathbb{N}$ such that $\{A\in\mathcal{A}:n\in A\}$ is uncountable.


EDIT: As seen in the solutions below, the passage to $\mathcal{A}\_{\subseteq}$ is unnecessary.


EDIT: In light of the positive answer to the questions above, I'll add an additional question that I am interested in.


Question 2: Must there be a *sequence* $(A\_n)\_{n\in\omega}$ of distinct elements of $\mathcal{A}$ (or elements of $\mathcal{A}\_{\subseteq}$ which are below pairwise distinct elements of $\mathcal{A}$) such that for any $n$, $\{B\in\mathcal{A}:\forall i\leq n(A\_i\upharpoonright n\subset B)\}$ (or $\{B\in\mathcal{A}:\forall i\leq n(A\_i\upharpoonright n-i\subset B)\}$) is uncountable. (Again, $A \upharpoonright n$ means the first $n$ elements of $A$ listed in increasing order, it does *not* mean the elements of $A$ below $n$ here.)



A property of uncountable almost disjoint families
Yes to both: Just take a condensation point $A$ of $\mathcal{A}$. This means every clopen neighborhood of $A$ contains uncountably many members of $\mathcal{A}$.


