A Mitchell-like order for Ramsey and Ramsey-like cardinals
Smallish large cardinals $\kappa $ are often characterized by the existence of a collection of filters on $\kappa $, each of which is an ultrafilter on the subsets of $\kappa $ of some transitive $\rm ZFC ^-$-model of size $ \kappa $. We introduce a Mitchell-like order for Ramsey and Ramsey-like cardinals, ordering such collections of small filters. We show that the Mitchell-like order and the resulting notion of rank have all the desirable properties of the Mitchell order on normal measures on a measurable cardinal. The Mitchell-like order behaves robustly with respect to forcing constructions. We show that extensions with the cover and approximation properties cannot increase the rank of a Ramsey or Ramsey-like cardinal. We use the results about extensions with the cover and approximation properties together with recently developed techniques about soft killing of large-cardinal degrees by forcing to softly kill the ranks of Ramsey and Ramsey-like cardinals.