Inverse limits of finite graphs

The random graph R is the unique countable graph which contains isomorphic copies of all countable graphs and is homogeneous in the sense that every isomorphism between its finite subgraphs extends to an automorphism of R. Actually, the existence, universality and uniqueness of the random graph follows from model-theoretic Fraisse theory. We study a projective counterpart of the random graph, that is, a graph represented as an inverse limit of finite graphs which would be projectively homogeneous with respect to finite graphs.