Let $L$ be a many-sorted first order language, and let $\kappa$ be an infinite cardinal which is greater than or equal to the number of function and relation symbols in $L$. Let $T$ be a complete theory over $L$ for which every sort is infinite. Let $S$ be the set of sorts of $L$ and let $Card_{\geq\kappa}$ be the class of cardinals which are $\geq\kappa$. Given a function $\pi:S\to Card_{\geq\kappa}$ (which I will call a *profile*), say that $\pi$ is $T$-*realizable* if there is a model $M$ of $T$ such that for each $s\in S$, $\pi(s)$ is the cardinality of the sort $s$ in the model $M$ (in which case I will say $\pi$ is the profile of $M$).

If there is only one sort, the Löwenheim-Skolem theorem says that every profile is realizable. However, when you have multiple sorts, there are theories for which not every profile is realizable. In particular, it is possible to have sorts $s$ and $t$ such that for every model, $|s|\leq|t|$ (e.g., if there is a definable injection from $s$ to $t$). More generally, if $F$ and $G$ are finite sets of sorts, it is possible that in every model, $\max_{s\in F} |s|\leq \max_{t\in G} |t|$ (e.g., if there is a definable injection from $\prod_F s$ to $\prod_G t$). More complicated kinds of restrictions are also possible (see Goldstern's comment below, for instance).

My question is:

What collections of functions $S\to Card_{\geq\kappa}$ can be the the collection of profiles of models of some theory? Given a particular theory $T$, is there a (relatively) easy way to determine which profiles are $T$-realizable?