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