I read on the Wikipedia page that "an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number." https://en.wikipedia.org/wiki/Equivalence_relation I understand the statement intuitively but I'm not sure if I'm on the right track proving it. Here is my approach to prove a theory with exactly two infinite equivalence classes is countably categorical but not uncountably categorical.
For every two countable models M1, M2 of the theory T of an equivalence relation with exactly two infinite equivalence classes. |M1|=|M2| since both equivalence classes have infinite elements. The domain of all models of T are countably infinite. Therefore there is an isomorphism between every two models of T.
For the second part, I'm trying to find a counterexample which shows two uncountable models are not isomorphic but I failed to find any. What I don't understand is, how can there be any uncountable model if the equivalence relation of a certain language has exactly two countably infinite equivalence classes?
Recall cardinal arithmetic for infinite cardinal: $$\kappa+\lambda=\max\{\kappa,\lambda\}.$$
Try to play with the cardinality of one of the two classes.