关于度量Ramsey型现象
On metric Ramsey-type phenomena
摘要：
本文研究的主要问题可以看作是巴拿赫空间理论中Dvoretzky定理的非线性模拟，也可以看作是组合学中拉姆齐理论的一部分。给定一个n点上的有限度量空间，我们寻找它的最大基数的子空间，该子空间可以嵌入希尔伯特空间中的给定的变形。我们提供了近紧的上限和下界的基数的这个子空间的n和所需的失真。我们的主要定理指出，对于任何n>0，每个n点度量空间包含一个大小至少为n^{1-ε}的子集，该子集可嵌入希尔伯特空间，失真为O（\frac{\log(1/ε)} ε）。失真的界限紧到log（1/ε）因子。我们还对这一问题的其他各个方面进行了全面研究。
英文摘要：
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-ε} which is embeddable in Hilbert space with O(\frac{\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.
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