Given a theory $T$ over a countable language with infinite models, and $\kappa$ an infinite cardinal, we can find a model of $T$ of size $\kappa$ whose infinite definable sets are all of size $\kappa$.
I have seen this claim in many lecture notes but have not been able to find one giving the actual proof. I only know of the compactness argument to find a model of size $\kappa$. I have read that the claim might follow from an application of the downward version of Löwenheim-Skolem's theorem, but I have no clue how to apply it here.
I could really use a hint. Thanks!