See Vasey 2014 for the first result, and Boney & Vasey 2014, Corollary 6.10.5 for the result on dimension. Vasey, Sebastien (2014). "Forking and superstability in tame AECs". arXiv:1405.7443v2 [math.LO]. Boney, Will; Vasey, Sebastien (2014). "Tameness and frames revisited". arXiv:1406.5980v4 [math.LO].
Lieberman 2011, Proposition 4.1. Lieberman, Michael J. (2011), "A topology for Galois types in abstract elementary classes", Mathematical Logic Quarterly, 57 (2): 204–216, doi:10.1002/malq.200910132