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 ǫ> 0, every n point metric space contains a subset of size at least n1−ǫ � which is embeddable in Hilbert space with O � log(1/ǫ) ǫ distortion. The bound on the distortion is tight up to the log(1/ǫ) factor. We further include a comprehensive study of various other aspects of this problem
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