Resumen de On metric Ramsey-type phenomena

Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor

• The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky¿s theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any
  > 0, every n point metric space contains a subset of size at least n1-
  which is embeddable in Hilbert space with O
  log(1/
  )
   distortion. The bound on the distortion is tight up to the log(1/
  ) factor. We further include a comprehensive study of various other aspects of this problem.