Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties

-  We define a notion of order of indiscernibility type of a structure by analogy with Mitchell order on measures; we use this to define a hierarchy of strong axioms of infinity defined through normal filters, the α-weakly Erdős hierarchy. The filters in this hierarchy can be seen to be generated by sets of ordinals where these indiscernibility orders on structures dominate the canonical functions.
-  The limit axiom of this is that of greatly Erdős and we use it to calibrate some strengthenings of the Chang property, one of which, CC+, is equiconsistent with a Ramsey cardinal, and implies that ω3=(ω2+)K where K is the core model built with non-overlapping extenders - if it is rigid, and others which are a little weaker. As one corollary we have: TheoremIf CC+∧¬□ω2then there is an inner model with a strong cardinal.
-  We define an α-Jónsson hierarchy to parallel the α-Ramsey hierarchy, and show that κ being α-Jónsson implies that it is α-Ramsey in the core model.