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)

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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 languageEnglish
Number of pages33
JournalarXiv
Publication statusPublished - 21 Apr 2015

Keywords

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

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