Saturation,Example, uncountable cardinality

Question

Here, in the example 1.0.27, how can I see that if $\lambda>\aleph_0$, then every model $M\in K_{\lambda^+}$ is saturated over $\lambda$?How do I use uncountability of $\lambda$?

Answer 1

You show it's saturated by figuring out what all the Galois types are (there aren't many in this example!), and checking that they're all realized.

Let's say an element which satisfies all the $P_n$ is central. The point is that if a model is uncountable of size $\lambda$, then it must contain a central element (actually $\lambda$-many), so you know which $E$-class all the central elements are in. Over a countable model with no central elements, there are two central Galois types, which can't both be realized in the same model. This is the only obstruction to saturation.