Computing the Number of Types of Infinite Length

Abstract

We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of $1$-types and the length of the sequences. Specifically, if $\kappa \le \lambda$, then

$${sup\hspace{0.17em}}_{\Vert M\Vert =\lambda }\left|S^{\kappa }\right(M\left)\right|=\left({sup\hspace{0.17em}}_{\Vert M\Vert =\lambda }\right|S^1\left(M\right)|{)}^{\kappa }.$$ We show that this holds for any abstract elementary class with $\lambda$-amalgamation. No such calculation is possible for nonalgebraic types. However, we introduce a subclass of nonalgebraic types for which the same upper bound holds.