Let $\kappa$ be an regular uncountable cardinal carrying a $\tau$-structure for some countable language $\tau$. What can be said regarding the existence of ordinals $\alpha <\kappa$ carrying elementary substructures of $\kappa$ ?
This is an (altered) question from an exercise which I am having difficulties solving. The Löwenheim–Skolem theorem doesn't seem to provide any direct insights here.