Random Geometric Graphs and Isometries of Normed Spaces

TL;DR

This paper investigates the uniqueness of Rado sets in finite-dimensional normed spaces, showing that only in $l__$ spaces do almost all dense sets produce isomorphic random geometric graphs, and characterizes spaces with Rado sets.

Contribution

It proves that $l__$ is the only finite-dimensional normed space where almost all dense sets are Rado, and characterizes spaces admitting Rado sets based on $l_$ summands.

Findings

$l__$ is unique in having almost all sets Rado.

Spaces with an $l_$ summand admit Rado sets.

Finite-dimensional spaces where step-isometries are isometries are characterized.

Abstract

Given a countable dense subset $S$ of a finite-dimensional normed space $X$, and $0<p<1$, we form a random graph on $S$ by joining, independently and with probability $p$, each pair of points at distance less than $1$. We say that $S$ is `Rado' if any two such random graphs are (almost surely) isomorphic. Bonato and Janssen showed that in $l_\infty^d$ almost all $S$ are Rado. Our main aim in this paper is to show that $l_\infty^d$ is the unique normed space with this property: indeed, in every other space almost all sets $S$ are non-Rado. We also determine which spaces admit some Rado set: this turns out to be the spaces that have an $l_\infty$ direct summand. These results answer questions of Bonato and Janssen. A key role is played by the determination of which finite-dimensional normed spaces have the property that every bijective step-isometry (meaning that the integer part of distances is preserved) is in fact an isometry. This result may be of independent interest.