Random graphs are not uncountably categorical

Question

Is there a simple proof that the theory of random graphs is not $\lambda$-categorical for uncountable $\lambda$?

Answer 1

Whether this counts as a simple proof or not will probably depend on your background.

An uncountably categorical countable theory is necessarily $\omega$-stable, which the random graph is very far from, which is easy to see directly. Alternatively, you may notice that no nonalgebraic type is stationary (but that requires a characterisation of forking, which does take some effort to establish).