For a first-order theory $T$ and cardinals $\kappa < \lambda$, we say that $M$ is a $(\kappa,\lambda)$-model if it is of size $\lambda$ and has a definable (with parameters) subset of size $\kappa$.
1) Let $T$ be the theory of the countable random graph. Which $(\kappa,\lambda)$-models does it admit?
2) For an arbitrary $T$, what are the sufficient conditions for the existence of $(\kappa,\lambda)$ models for some $\kappa < \lambda$? This is not a question about transfer from some $(\kappa,\lambda)$ to a different $(\kappa',\lambda')$, there are quite a few theorems there. What I am asking for is some kind of a non-structure theorem, (apart from having a Vaughtian pair).