Recursively saturated, resplendent models, and saturated models.
From Peano's Parlour
Omitting theories of undefinable sets
Suppose M is countable recursively saturated and X is an undefinable subset of M. Is there a countable recursively saturated N such that N is an elementary end extension of M, and if Y⊆M is coded in N, then (M,Y)≢(M,X)?
The answer if `yes' is either (M,X)⊭PA∗ or Th(M,X)∉SSy(M).
This problem is listed in Kossak, Roman; Schmerl, James H. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006. xiv+311, but, unfortunately, with many typos.