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We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated "algebraic closure" operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases.

We consider tame minimal simple groups of finite Morley rank and of odd type. We show that the Prüfer 2-rank of such a group is bounded by 2. We also find all potential nonalgebraic configurations; there are essentially four of them, and we delineate them with some precision.

We clarify quasi-Frobenius configurations of finite Morley rank. 1. We remove one assumption in an identification theorem by Zamour while simplifying the proof. 2. We show that a strongly embedded quasi-Frobenius configuration of odd type, is actually Frobenius. 3. For dihedral configurations, one has dim G = 3 dim C. These results rely on an interesting phenomenon of closure of non-generic matter under taking centralisers.