TITLE: Ideals on $\mathbb N$ and large sets that have small intersection
QUESTION [8 upvotes]: Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable:
$$A,B\in \mathcal A, A\neq B \Rightarrow A\cap B \in \mathcal I. $$
Let us observe that maximal ideals have this property as only the empty family or the singletons meet this requirement.

Are there further examples of ideals with this property? 

(As observed by Leonetti every such ideal must be non-meagre when regarded as a subset of the Cantor set.)

REPLY [3 votes]: In topological language: for any closed subset, $F$, of $\beta\mathbb{N}\setminus\mathbb{N}$ the family $I_F=\{A:A^*\cap F=\emptyset\}$ is an ideal.
Your property translates into: the closed set $F$ is a ccc space.
The example by Monk can be made in this way too: take a map $s$ from $\beta\mathbb{N}\setminus\mathbb{N}$ onto the Cantor set $C$ and take $F$ such that $s:F\to C$ is irreducible and onto, then $F$ is sparable and hence ccc.
The example by Blass corresponds to the closure of a countable set in $\beta\mathbb{N}\setminus\mathbb{N}$, again a separable subspace.
In general: take any compact ccc space, $X$, of weight at most $\mathfrak{c}$ and embed it into the Tychonoff cube $[0,1]^\mathfrak{c}$. There is a continuous map $\sigma$ from $\beta\mathbb{N}$ onto that cube. Now take $F$ such that $\sigma:F\to X$ is irreducible and onto; then $F$ is ccc. For a non-separable example let $X$ be the Stone space of the measure algebra.