On metric Ramsey-type phenomena
<|reference_start|>On metric Ramsey-type phenomena: 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.<|reference_end|>
@article{bartal2004on,
    title={On metric Ramsey-type phenomena},
    author={Yair Bartal, Nathan Linial, Manor Mendel, Assaf Naor},
    journal={Ann. of Math. (2) 162 (2005), no. 2, 643--709},
    year={2004},
    doi={10.4007/annals.2005.162.643},
    archivePrefix={arXiv},
    eprint={math/0406353},
    primaryClass={math.MG cs.DS}
}