Random geometric graphs and isometries of normed spaces

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 $\ell _\infty ^d$ almost all $S$ are Rado. Our main aim in this paper is to show that $\ell _\infty ^d$ 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 $\ell _\infty$ 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.