  Recently, connections have been explored between the complexity of finite
problems in graph theory and the complexity of their infinite counterparts. As
is shown in our paper (and in independent work of Tirza Hirst and D. Harel from
a different angle) there is no firm connection between these complexities,
namely finite problems of equal complexity can have radically different
complexity for the infinite versions and vice versa. Furthermore, the
complexity of an infinite counterpart can depend heavily on precisely how the
finite problem is rephrased in the infinite case.
  The finite problems we address include colorability of graphs and existence
of subgraph isomorphisms. In particular, we give three infinite versions of the
3-colorability problem that vary considerably in their recursion theoretic and
proof theoretic complexity. Additionally, we show that three subgraph
isomorphism problems of varying finite complexity have infinite versions with
identical recursion theoretic and proof theoretic content.
