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 prove that for C a finite set of cycles, there is a universal C-free graph if and only if C consists precisely of all the odd cycles of order less than same specified bound.The sufficiency of this condition was proved by Komjath, Mekler, and Pach (Israel J.
The problem of the existence of a universal structure omitting a finite set of forbidden substructures is reducible to the corresponding problem in the category of graphs with a vertex coloring by two colors. It is not known whether this problem reduces further to the category of ordinary graphs. It is also not known whether these problems are decidable.
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