On metric ramsey-type phenomena

  • Authors:

  • Yair Bartal;Nathan Linial;Manor Mendel;Assaf Naor

  • Affiliations:

  • Hebrew University, Jerusalem, Israel;Hebrew University, Jerusalem, Israel;Hebrew University, Jerusalem, Israel;Theory Group, Microsoft Research, Redmond, WA

  • Venue:
  • Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
  • Year:
  • 2003

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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 ε0, every n point metric space contains a subspace of size at least n1-ε which is embeddable in an ultrametric with O(log(1/ε)/ε distortion. 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.