One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L.
Victoria Gitman. "Ramsey-like cardinals." J. Symbolic Logic 76 (2) 519 - 540, June 2011. https://doi.org/10.2178/jsl/1305810762