### 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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Pages (from-to) | 463-472 |

Number of pages | 10 |

Journal | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

DOIs | |

State | Published - 2003 |

Externally published | Yes |

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

### All Science Journal Classification (ASJC) codes

- Software

### Keywords

- Dvoretzky theorem
- Finite metric spaces
- Ramsey theory

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## Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*, 463-472. https://doi.org/10.1145/780542.780610