Geometric random graphs and Rado sets in sequence spaces

Abstract

We consider a random geometric graph model, where pairs of vertices are points in a metric space and edges are formed independently with fixed probability p between pairs within threshold distance 1. A countable dense set in a metric space is Rado if this random model gives, with probability 1, a graph that is unique up to isomorphism. In earlier work, the first two authors proved that, in finite dimensional spaces Rn equipped with the ℓ∞ norm, all countable dense sets satisfying a mild non-integrality condition are Rado. In this paper we extend this result to two infinite-dimensional spaces: c and c0. If the underlying metric space is a separable Banach space, then we show in some cases that we can almost surely recover the Banach space from such a geometric random graph. More precisely, we show that in the sequence spaces c and c0, for measures μ satisfying certain conditions, μN-almost all countable sets are Rado. Moreover, with probability 1, in c as in c0, all graphs obtained from the random geometric model with a randomly chosen dense countable vertex set are isomorphic to each other. Finally, we show that representatives of the isomorphism classes obtained in this way from c and c0 are non-isomorphic to each other, and also non-isomorphic to their counterparts obtained from finite dimensional spaces.