Let $\kappa$ be a measurable cardinal. We give the Stone space of all ultrafilters on $\kappa$ the usual topology, where each $x\subseteq\kappa$ determines a basic open $[x]=\{U;x\in U\}$. The subspace $m(\kappa)$ of all normal measures on $\kappa$ inherits a topology.
Given a discrete subspace $A\subseteq m(\kappa)$, a discretizing family for $A$ is a family of subsets of $\kappa$ witnessing that $A$ is discrete, i.e. a family $\{x_U;U\in A\}$ such that each $U$ is the unique element of $A$ with $x_U\in U$.
I am interested in the sizes of discrete subspaces of $m(\kappa)$ with nice discretizing families. It is a straightforward exercise that any subset of $m(\kappa)$ of size at most $\kappa$ is discrete and, moreover, has an almost disjoint discretizing family (meaning that all pairwise intersections of the elements of the family are bounded in $\kappa$). By (possibly) passing to a forcing extension we can push this to a (consistencywise) optimal result: a measurable Laver function for $\kappa$ yields a discrete set of measures of size $2^\kappa$ (clearly the largest possible) with an almost disjoint discretizing family.
On the other hand, if the Mitchell rank of $\kappa$ is at least $\kappa^+$ then there is a discrete subspace of size $\kappa^+$ with a discretizing family whose pairwise intersections have measure 0 with respect to any normal measure on $\kappa$.
I would be interested in knowing whether any of these distinctions are strict. Specifically:
Is it consistent that $m(\kappa)$ has a discrete subspace of size $\kappa^+$ with a discretizing family with "not too big" pairwise intersections (such as the absolute measure 0 example above), but no such subspace with an almost disjoint discretizing family? In particular, does this happen in the canonical model $L[\vec{U}]$ for $o(\kappa)=\kappa^+$?
Is it consistent that $\kappa$ has at least $\kappa^+$ many normal measures but there are no discrete subspaces of $m(\kappa)$ of size $\kappa^+$?
Question 2 in particular seems quite difficult, since it fundamentally relies on controlling the number of normal measures at $\kappa$. All of the forcing methods I am aware of that achieve this, starting from a measurable, inadvertently add a measurable Laver function, which, as mentioned above, produces a large discrete set of measures with an almost disjoint discretizing family.
