Counting partial types in simple theories

We continue the work of Shelah and Casanovas on the cardinality of families of pairwise inconsistent types in simple theories. We prove that, in a simple theory, there are at most ${\displaystyle {\lambda }^{<\kappa \left(T\right)}+2^{\mu +\left|T\right|}}$ pairwise inconsistent types of size μ over a set of size λ. This bound improves the previous bounds and clarifies the role of κ(T). We also compute exactly the maximal cardinality of such families for countable, simple theories. The main tool is the fact that, in simple theories, the collection of nonforking extensions of fixed size of a given complete type (ordered by reverse inclusion) has a chain condition. We show also that for a notion of dependence, this fact is equivalent to Kim-Pillay's type amalgamation theorem; a theory is simple if and only if it admits a notion of dependence with this chain condition, and furthermore that notion of dependence is forking.