2023/03/20 by Ben-Neria, Omer, Kaplan, Itay, Zou, Tingxiang
#03C45 #03C52 #03C64 #05C80 #FOS: Mathematics #Logic (math.LO) #Probability (math.PR)
paper · doi:10.48550/arxiv.2303.11292
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.