On metric Ramsey-type phenomena
摘要
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.
引用
@article{arxiv.math/0406353,
title = {On metric Ramsey-type phenomena},
author = {Yair Bartal and Nathan Linial and Manor Mendel and Assaf Naor},
journal= {arXiv preprint arXiv:math/0406353},
year = {2012}
}
备注
67 pages, published version