Some low distortion metric ramsey problems

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

Research output: Contribution to journalArticle

Abstract

In this note we consider the metric Ramsey problem for the normed spaces ℓp. Namely, given some 1 ≤ p ≤ ∞ and α ≥ 1, and an integer n, we ask for the largest m such that every n-point metric space contains an m-point subspace which embeds into ℓp with distortion at most α. In [1] it is shown that in the case of ℓ2, the dependence of m on α undergoes a phase transition at α = 2. Here we consider this problem for other ℓp, and specifically the occurrence of a phase transition for p ≠ 2. It is shown that a phase transition does occur at α = 2 for every p ∈ [1, 2]. For p > 2 we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every 1 < p < ∞ there are arbitrarily large metric spaces, no four points of which embed isometrically in ℓp.

Original languageEnglish (US)
Pages (from-to)27-41
Number of pages15
JournalDiscrete and Computational Geometry
Volume33
Issue number1
DOIs
StatePublished - 2005

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Phase Transition
Phase transitions
Metric
Metric space
Isometric Embedding
Normed Space
Subspace
Integer
Estimate

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

Some low distortion metric ramsey problems. / Bartal, Yair; Linial, Nathan; Mendel, Manor; Naor, Assaf.

In: Discrete and Computational Geometry, Vol. 33, No. 1, 2005, p. 27-41.

Research output: Contribution to journalArticle

Bartal, Y, Linial, N, Mendel, M & Naor, A 2005, 'Some low distortion metric ramsey problems', Discrete and Computational Geometry, vol. 33, no. 1, pp. 27-41. https://doi.org/10.1007/s00454-004-1100-z
Bartal, Yair ; Linial, Nathan ; Mendel, Manor ; Naor, Assaf. / Some low distortion metric ramsey problems. In: Discrete and Computational Geometry. 2005 ; Vol. 33, No. 1. pp. 27-41.
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