A Geometric Zero-One Law

187 – Each relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. If x is an element of X, let B n (x) be the ball of radius n around x. Suppose that X is infinite, connected and of bounded degree. A first-order sentence s in the language of X is almost surely true (resp. a.s. false) for finite substructures of X if for every x in X, the fraction of substructures of B n (x) satisfying s approaches 1 (resp. 0) as n approaches infinity. Suppose further that, for every finite substructure, X has a disjoint isomorphic substructure. Then every s 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.