  We study PC-exact saturation for stable and simple theories. Among other
results, we show that PC-exact saturation characterizes the stability cardinals
of size at least continuum of a countable stable theory and, additionally, that
simple unstable theories have PC-exact saturation at singular cardinals,
satisfying mild set-theoretic hypotheses, which had previously been open even
for the random graph. We characterize supersimplicity of countable theories in
terms of having PC-exact saturation at singular cardinals of countable
cofinality. We also consider the local analogue of PC-exact saturation, showing
that local PC-exact saturation for singular cardinals of countable cofinality
characterizes supershort theories.
