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.



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



Ideals on $\mathbb N$ and large sets that have small intersection
The denumerable atomless Boolean algebra $A$ can be isomorphically embedded in $\wp(\omega)/\mathsf{fin}$. The identity on $A$ can be extended to a homomorphism from $\wp(\omega)/\mathsf{fin}$ into the completion of $A$. This gives a homomorphic image of $\wp(\omega)/\mathsf{fin}$, hence also of $\wp(\omega)$ itself, in which every disjoint subset is countable. So the kernel of the homomorphism is another ideal of the type indicated.


