Abstract
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 $O\left(\frac{\log(1/\epsilon)}{\epsilon}\right)$ 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.
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