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arXiv · 2303.11292

The model theory of geometric random graphs

Omer Ben-Neria·Itay Kaplan·Tingxiang Zou
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Abstract

We study the logical properties of infinite geometric random graphs, introduced by Bonato and Janssen. These are graphs whose vertex set is a dense ``generic'' subset of a metric space, where two vertices are adjacent with probability $p>0$ provided the distance between them is bounded by some constant number. We prove that for a large class of metric spaces, including circles, spheres and the complete Urysohn space, almost all geometric random graphs on a given space are elementary equivalent. Moreover, their first-order theory can reveal geometric properties of the underlying metric space.

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Omer Ben-Neria, Itay Kaplan, Tingxiang Zou. 2023-03-20. The model theory of geometric random graphs. https://arxiv.org/abs/2303.11292

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