$\kappa$-stable theories and number of types

Question

How can I show that if $T$ is a $\kappa$-stable theory, then in each model of $T$, over every set of parameters with at most $\kappa$ many elements, there are at most $\kappa$ many n–types.

Answer 1

I'm assuming that your definition of $\kappa$-stable is "In each model of $T$, over every set of parameters with at most $\kappa$ many elements, there are at most $\kappa$ many $1$-types".

This is the base case of an induction on $n$. Hint for the inductive step: The $(n+1)$-type $\text{tp}(a_0,a_1,\dots,a_n \,/\, B)$ is determined by the $n$-type $\text{tp}(a_1,\dots,a_n\,/\,B)$ and the $1$-type $\text{tp}(a_0\,/\,B\cup\{a_1,\dots,a_n\})$.