In the book Baldwin, Categoricity in Abstract Elementary Classes defines (Def.20.1,p.151) a notion of Tarski-Vaught extensions for tuples that generalises both independence and usual Tarski-Vaught extensions (Ex 20.4,p.152).

Is there any "finite character" for Tarski-Vaught for tuples generalising finite character for independence(splitting) ? That is, a generalisation of Exercise 19.15,p.147 ?

I would also be interested to see a proof of equivalence of definitions of a *good system* as defined by Baldwin and by Shelah in the original paper.