Aspects of stability in simple theories

Simple theories are a strict extension of stable theories for which non-forking independence is a nice independence relation. However, not much is known about how the simple unstable theories differ from the strictly stable ones. This work looks at three aspects of simple theories and uses them to give a better picture of the differences between the two classes. First, we look at the property of weakly eliminating hyperimaginaries and show that it is equivalent to forking and thorn-forking independence coinciding. Second, we look at the stable forking conjecture}, a strong statement asserting that simple unstable theories have an essentially stable 'core,' and prove that it holds between elements having SU-rank 2 and finite SU-rank. Third, we consider a property on indiscernible sequences that is known to hold in every stable theory, and show it holds on, at most, a subset of simple theories out of all possible first order theories.