Source: J. Symbolic Logic
Volume 74, Issue 3
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 Bn(x) be the ball of radius n around x. Suppose that X is
infinite, connected and of bounded degree. A first-order sentence φ
in the language of X is almost surely true (resp. a.s. false) for
finite substructures of X if for every x∈ X, the fraction of
substructures of Bn(x) satisfying φ approaches 1 (resp. 0) as
n approaches infinity. Suppose further that, for every finite
substructure, X has a disjoint isomorphic substructure. Then every
φ 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.
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