Abstract: This chapter makes use of an induced version of Ramsey's Theorem to prove several results: the Abramson-Harrington theorem on omitting large indiscernible sets and Hanf numbers for models of arithmetic; a theorem characterizing the possible automorphism groups of models; and a theorem based on countable recursively saturated models being generated by a set of indiscernibles.

Keywords: Ramsey's Theorem, Abramson-Harrington theorem, Hanf numbers, automorphism groups, indiscernibles,