This paper continues the study of the Ramsey-like large cardinals introduced in  and . Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α≤ω₁, that they are downward absolute to L for α <ω₁L, and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.
We show that the strongly Ramsey and super Ramsey cardinals from  are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in  as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.
"Ramsey-like cardinals II." J. Symbolic Logic 76 (2) 541 - 560, June 2011. https://doi.org/10.2178/jsl/1305810763