Universal graphs with forbidden subgraphs and algebraic closure
  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. 
Gregory Cherlin, Saharon Shelah, Niandong Shi