Geometric random graphs and Rado sets of continuous functions

TL;DR

This paper establishes the existence of Rado sets in the space of continuous functions, showing that most dense sets in certain subspaces generate unique, isomorphic geometric random graphs, highlighting structural distinctions among different function spaces.

Contribution

It proves the existence of Rado sets in continuous function spaces and characterizes the isomorphism types of the resulting geometric random graphs for various subspaces.

Findings

Almost all dense sets in piecewise linear functions are Rado.

Almost all Brownian motion path sets are Rado.

Graphs from different subspaces are non-isomorphic.

Abstract

We prove the existence of Rado sets in the Banach space of continuous functions on [0,1]. A countable dense set S is Rado if with probability 1, the infinite geometric random graph on S, formed by probabilistically making adjacent elements of S that are within unit distance of each other, is unique up to isomorphism. We show that for a suitable measure which we construct, almost all countable dense sets in the subspaces of piecewise linear functions and of polynomials are Rado. Moreover, all graphs arising from such sets are of a unique isomorphism type. For the subspace of Brownian motion paths, almost all countable subsets are Rado (for a suitable measure) and the resulting graphs are of a unique isomorphism type. We show that the graph arising from piecewise linear functions and polynomials is not isomorphic to the graph arising from Brownian motion paths. Moreover, these graphs are non-isomorphic to graphs arising from Rado sets in $\mathbb{R}^n$, or the sequence spaces $c$ and $c_0$.