Small Extensions of Models of $o$-Minimal Theories and Absolute Homogeneity

We obtain some results on existence of small extensions of models of weakly $o$-minimal atomic theories. In particular, we find a sharp upper estimate for the Hanf number of such a theory for omitting an arbitrary family of pure types. We also find a sharp upper estimate for cardinalities of weakly $o$-minimal absolutely homogeneous models and a sufficient condition for absolute homogeneity.