On metric ramsey-type phenomena

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. Contents

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor
• Keywords: Algorithms, Banach Space Theory, Finite Metric Space, Point Metric Space, STOC 2003