Infinite Random Geometric Graphs.

We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability $${p \in (0, 1)}$$ if the metric distance between the vertices is below a given threshold. For certain choices of V as a countable dense set in $${\mathbb{R}^n}$$ equipped with the metric derived from the L-norm, it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR, is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GR. In contrast, we show that infinite random geometric graphs in $${\mathbb{R}^{2}}$$ with the Euclidean metric are not necessarily isomorphic. [ABSTRACT FROM AUTHOR]