Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note that some texts also assume that $M$ must be uncountable).
Note that if we leave out the size condition, then the theory of arithmetic has a Jónsson model of size $\aleph_0$. This model is $(\mathbb{N};0,\leq,+,\times)$, i.e. the standard natural numbers. Note that this model is trivially Jónsson since any proper subset is not a model of arithmetic (which is what happens in the countable case for all theories).
I was curious if anyone had any examples of theories with uncountable Jónsson models and what explicitly those models look like (the only posts I could find were about what theories never have uncountable Jónsson models, e.g. atomless Boolean algebras).
Thanks!