The Complexity of Infinite H-Coloring

For a fixed graph H, the homomorphism problem for H is the problem of determining whether or not there is a homomorphism of a finite input graph G into H. We investigate the complexity of this problem when H is allowed to be countably infinite and show that there exist recursive graphs with unsolvable homomorphism problems, as well as recursive graphs with solvable homomorphism problems of very high complexity. In fact, we show that there exist graphs with arbitrary recursively enumerable degrees of unsolvability.