TITLE: Cardinality of families of subsets of $\mathbb{N}$ whose intersections are finite
QUESTION [11 upvotes]: Does there exist an uncountable $P \subset \mathcal{P}(\mathbb{N}) $ with the property that for any distinct $x,y \in P$, $|x \cap y|$ is prime? 
A more general, but likely harder, question: is it possible to characterize the set $\mathcal{A}$ of all subsets $S \subset \mathbb{N}$ with the following property: there is some uncountable $P \subset \mathcal{P}(\mathbb{N})$ such that for all distinct $x,y \in P$, $|x \cap y| \in S$. 
It's easy to show no $S$ finite belongs to $\mathcal{A}$. It's also not hard to show every $\text{mod} \  m$ equivalence class, belongs to $\mathcal{A}$.
One reasonable sounding idea is that this is related to the asymptotic density of $S$ in $\mathbb{N}$.

REPLY [2 votes]: The following argument uses less set-theory than the earlier ones!
For each positive real number $\alpha$ we can write the binary expansion $\alpha=\sum_{i=0}^{\infty} a_i 2^{N-i}$ with $a_0=1$ and $a_i\in\{0,1\}$ for $i\geq 1$. To avoid ambiguity (from an endless sequence of 1's) we also assume that for all $j$ there is a $k>j$ so that $a_k=0$.
We take $S(\alpha)=\{ \alpha_n : n>0 \}$ where $\alpha_n=\sum_{i=0}^n a_i 2^{N-i}$. Then $S(\alpha)\subset\mathbb{Q}$, so we have an uncountable collection of subsets of $\mathbb{Q}$ which is in bijection with $\mathbb{N}$. Moreover, $S(\alpha)\cap S(\beta)$ is finite when $\alpha\neq\beta$. (It consists of those truncated binary expansions of $\alpha$ and $\beta$ which are equal.)
Now take the subset of reals which consists of those $\alpha$ for which $a_i=1$ unless $i$ is prime. It is not too difficult to see that these $\alpha$ are also uncountable. For example you can map reals to such numbers by "filling in 1's" in the binary expansion as required so that the old $a_i$'s become the new $a_{\pi(i)}$ where $\pi(n)$ is the $n$-th prime.
Moreover, it follows that $S(\alpha)\cap S(\beta)$ is a prime for every such $\alpha$ and $\beta$.