On metric Ramsey-type phenomena

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

Research output: Contribution to journalConference article

34 Scopus citations

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)
Pages (from-to)463-472
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs
StatePublished - 2003
Externally publishedYes
Event35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: Jun 9 2003Jun 11 2003

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Dvoretzky theorem
  • Finite metric spaces
  • Ramsey theory

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