Exact saturation in pseudo-elementary classes for simple and stable theories

We use exact saturation to study the complexity of unstable theories, showing that a variant this notion called pseudo-elementary class (PC)-exact meaningfully reflects combinatorial dividing lines. PC-exact for stable and simple theories. Among other results, we show characterizes stability cardinals size at least continuum countable theory and, additionally, theories have singular satisfying mild set-theoretic hypotheses. This had previously been open even random graph. characterize supersimplicity in terms having cofinality. also consider local analog saturation, cofinality supershort