On metric ramsey-type dichotomies

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

Research output: Contribution to journalArticle

Abstract

The classical Ramsey theorem states that every graph contains either a large clique or a large independent set. Here similar dichotomic phenomena are investigated in the context of finite metric spaces. Namely, statements are provided of the form 'every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral'. Two distinct interpretations are considered for being 'far from equilateral'. Proximity among metric spaces is quantified through the metric distortion α. Tight asymptotic answers are provided for these problems. In particular, it is shown that a phase transition occurs at α = 2.

Original languageEnglish (US)
Pages (from-to)289-303
Number of pages15
JournalJournal of the London Mathematical Society
Volume71
Issue number2
DOIs
StatePublished - Apr 2005

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Equilateral
Dichotomy
Metric space
Metric
Ramsey's Theorem
Independent Set
Clique
Large Set
Proximity
Phase Transition
Subspace
Distinct
Graph in graph theory

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On metric ramsey-type dichotomies. / Bartal, Yair; Linial, Nathan; Mendel, Manor; Naor, Assaf.

In: Journal of the London Mathematical Society, Vol. 71, No. 2, 04.2005, p. 289-303.

Research output: Contribution to journalArticle

Bartal, Y, Linial, N, Mendel, M & Naor, A 2005, 'On metric ramsey-type dichotomies', Journal of the London Mathematical Society, vol. 71, no. 2, pp. 289-303. https://doi.org/10.1112/S0024610704006155
Bartal, Yair ; Linial, Nathan ; Mendel, Manor ; Naor, Assaf. / On metric ramsey-type dichotomies. In: Journal of the London Mathematical Society. 2005 ; Vol. 71, No. 2. pp. 289-303.
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