Problem of finite axiomatizability for strongly minimal theories of graphs and groups with a nonzero number of ends (Q2639054)

The main result is the following: Let T be a strongly minimal theory of graphs of bounded valence with a nonzero number of ends and T be not categorical in countable power. Then T is not finitely axiomatizable modulo positive propositions. This implies, in particular, that any finitely generated infinite group such that its graph has a nonzero number of ends does not contain a finite set of classes of conjugate elements such that each nontrivial cyclic subgroup has a nontrivial intersection with one of these classes.