### 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 language | English (US) |
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Pages (from-to) | 289-303 |

Number of pages | 15 |

Journal | Journal of the London Mathematical Society |

Volume | 71 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2005 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of the London Mathematical Society*,

*71*(2), 289-303. https://doi.org/10.1112/S0024610704006155

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

Research output: Contribution to journal › Article

*Journal of the London Mathematical Society*, vol. 71, no. 2, pp. 289-303. https://doi.org/10.1112/S0024610704006155

}

TY - JOUR

T1 - On metric ramsey-type dichotomies

AU - Bartal, Yair

AU - Linial, Nathan

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2005/4

Y1 - 2005/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33644613359&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644613359&partnerID=8YFLogxK

U2 - 10.1112/S0024610704006155

DO - 10.1112/S0024610704006155

M3 - Article

VL - 71

SP - 289

EP - 303

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 2

ER -