On metric Ramsey-type phenomena

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 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.

Original languageEnglish (US)
Title of host publicationConference Proceedings of the Annual ACM Symposium on Theory of Computing
Pages463-472
Number of pages10
StatePublished - 2003
Event35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: Jun 9 2003Jun 11 2003

Other

Other35th Annual ACM Symposium on Theory of Computing
CountryUnited States
CitySan Diego, CA
Period6/9/036/11/03

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Banach spaces
Geometry

Keywords

  • Dvoretzky theorem
  • Finite metric spaces
  • Ramsey theory

ASJC Scopus subject areas

  • Software

Cite this

Bartal, Y., Linial, N., Mendel, M., & Naor, A. (2003). On metric Ramsey-type phenomena. In 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.

Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2003. p. 463-472.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Bartal, Y, Linial, N, Mendel, M & Naor, A 2003, On metric Ramsey-type phenomena. in 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.
Bartal Y, Linial N, Mendel M, Naor A. On metric Ramsey-type phenomena. In Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2003. p. 463-472
Bartal, Yair ; Linial, Nathan ; Mendel, Manor ; Naor, Assaf. / On metric Ramsey-type phenomena. Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2003. pp. 463-472
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