A Geometric Zero-One Law

TL;DR

This paper introduces a geometric zero-one law for infinite, connected, bounded degree structures, showing that certain first-order properties are almost surely true or false in finite substructures, and explores related questions.

Contribution

It formulates a new geometric zero-one law applicable to infinite structures with specific properties and investigates its implications without referencing the ambient structure.

Findings

Every first-order sentence is almost surely true or false under the law.

The law applies to structures with disjoint isomorphic substructures.

The paper explores various questions related to the law's properties.

Abstract

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.