# Random Geometric Graphs and Isometries of Normed Spaces

Paul Balister, Béla Bollobás, Karen Gunderson, Imre Leader, Mark Walters

Research output: Contribution to journalArticle (Academic Journal)

## Abstract

Given a countable dense subset $S$ of a finite-dimensional normed space $X$, and $0p`Rado if any two such random graphs are (almost surely) isomorphic. Bonato and Janssen showed that in$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.
Original language English 33 arXiv Published - 21 Apr 2015

• math.FA
• math.CO
• 05C63
• 05C80
• 46B04

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