Let $n\in\mathbb{N}$ be fixed. Given a family $\mathcal{F}$ of subsets of $\mathbb{N}$, each of size $n$, with $\bigcup\mathcal{F}=\mathbb{N}$, set
$$d(\mathcal{F}):=\max\{k\in\mathbb{N}:\;\exists\,\text{partition}\,\mathbb{N}=N_1\cup\ldots\cup N_k\;\text{s.t.}\;\forall j=1,\ldots, k: N_j\cap F\neq\emptyset\;\forall F\in\mathcal{F}\}$$
This is well-defined since $k=1$ is always in the set and clearly $d(\mathcal{F})\leq n$. It is some sort of disjointness "measure" because the upper bound is attained in particular by those $\mathcal{F}$, which are partitions of $\mathbb{N}$. The families $\mathcal{F}$ can be classified by $d(\mathcal{F})=k$ for $k=1,\ldots, n$.
I'm looking for a name of this concept and in particular I'm interested in what the families in each class look like. There must be something along these lines in literature as the whole thing can be restated in terms of multi-partite stuff in uniform hypergraphs. Thanks (for any pointers)!