If $\kappa$ is an infinite cardinal, every maximal diverse family of subsets of $\kappa$ has cardinality $2^\kappa$.

**Proof.** Let me use the usual notation $X\operatorname{\triangle}Y=(X\setminus Y)\cup(Y\setminus X)$. The relation $|X\operatorname{\triangle}Y|\lt\kappa$ is an equivalence relation. A "maximal diverse family" $\mathcal E$ is just a set of representatives for that equivalence relation, so the cardinality of $\mathcal E$ is just the number of equivalence classes, which is $2^\kappa$.