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