  Each relational structure X has an associated Gaifman graph, which endows X
with the properties of a graph. Suppose that X is infinite, connected and of
bounded degree. A first-order sentence in the language of X is almost surely
true (resp. a.s. false) for finite substructures of X if for every element x in
X, the fraction of substructures of the ball of radius n around x which satisfy
the sentence approaches 1 (resp. 0) as n approaches infinity. Suppose further
that, for every finite substructure, X has a disjoint isomorphic substructure.
Then every sentence is a.s. true or a.s. false for finite substructures of X.
This is one form of the geometric zero-one law. We formulate it also in a form
that does not mention the ambient infinite structure. In addition, we
investigate various questions related to the geometric zero-one law.
