### Abstract

This paper deals with Ramsey-type theorems for metric spaces. Such a theorem states that every n point metric space contains a large subspace which can be embedded with some fixed distortion in a metric space from some special class. Our main theorem states that for any ε > 0, every n point metric space contains a subspace of size at least n^{1-ε}which is embeddable in an ultrumetric with O(log(1/e/e) dis-tortion. This in particular provides a bound for embedding in Euclidean spaces. The bound on the distortion is tight up to the log(1/ε) factor even for embedding in arbitrary Euclidean spaces. This result can be viewed as a non-linear analog of Dvoretzky's theorem, a cornerstone of modern Banach space theory and convex geometry. Our main Ramsey-type theorem and techniques naturally extend to give theorems for classes of hierarchically well-separated trees which have algorithmic implications, and can be viewed as the solution of a natural clustering problem. We further include a comprehensive study of various other aspects of the metric Ramsey problem.

Original language | English (US) |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Pages | 463-472 |

Number of pages | 10 |

State | Published - 2003 |

Event | 35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: Jun 9 2003 → Jun 11 2003 |

### Other

Other | 35th Annual ACM Symposium on Theory of Computing |
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Country | United States |

City | San Diego, CA |

Period | 6/9/03 → 6/11/03 |

### Fingerprint

### Keywords

- Dvoretzky theorem
- Finite metric spaces
- Ramsey theory

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*(pp. 463-472)

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing.*pp. 463-472, 35th Annual ACM Symposium on Theory of Computing, San Diego, CA, United States, 6/9/03.

}

TY - GEN

T1 - On metric Ramsey-type phenomena

AU - Bartal, Yair

AU - Linial, Nathan

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2003

Y1 - 2003

N2 - This paper deals with Ramsey-type theorems for metric spaces. Such a theorem states that every n point metric space contains a large subspace which can be embedded with some fixed distortion in a metric space from some special class. Our main theorem states that for any ε > 0, every n point metric space contains a subspace of size at least n1-εwhich is embeddable in an ultrumetric with O(log(1/e/e) dis-tortion. This in particular provides a bound for embedding in Euclidean spaces. The bound on the distortion is tight up to the log(1/ε) factor even for embedding in arbitrary Euclidean spaces. This result can be viewed as a non-linear analog of Dvoretzky's theorem, a cornerstone of modern Banach space theory and convex geometry. Our main Ramsey-type theorem and techniques naturally extend to give theorems for classes of hierarchically well-separated trees which have algorithmic implications, and can be viewed as the solution of a natural clustering problem. We further include a comprehensive study of various other aspects of the metric Ramsey problem.

AB - This paper deals with Ramsey-type theorems for metric spaces. Such a theorem states that every n point metric space contains a large subspace which can be embedded with some fixed distortion in a metric space from some special class. Our main theorem states that for any ε > 0, every n point metric space contains a subspace of size at least n1-εwhich is embeddable in an ultrumetric with O(log(1/e/e) dis-tortion. This in particular provides a bound for embedding in Euclidean spaces. The bound on the distortion is tight up to the log(1/ε) factor even for embedding in arbitrary Euclidean spaces. This result can be viewed as a non-linear analog of Dvoretzky's theorem, a cornerstone of modern Banach space theory and convex geometry. Our main Ramsey-type theorem and techniques naturally extend to give theorems for classes of hierarchically well-separated trees which have algorithmic implications, and can be viewed as the solution of a natural clustering problem. We further include a comprehensive study of various other aspects of the metric Ramsey problem.

KW - Dvoretzky theorem

KW - Finite metric spaces

KW - Ramsey theory

UR - http://www.scopus.com/inward/record.url?scp=0038446933&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038446933&partnerID=8YFLogxK

M3 - Conference contribution

SP - 463

EP - 472

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

ER -