TY - JOUR

T1 - On metric Ramsey-type phenomena

AU - Bartal, Yair

AU - Linial, Nathan

AU - Mendel, Manor

AU - Naor, Assaf

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2005/9

Y1 - 2005/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=28944451802&partnerID=8YFLogxK

U2 - 10.4007/annals.2005.162.643

DO - 10.4007/annals.2005.162.643

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AN - SCOPUS:28944451802

SN - 0003-486X

VL - 162

SP - 643

EP - 709

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -