度量Ramsey理论及其应用的进展
数据结构与算法
2021-04-09 v1 计算几何
度量几何
摘要
度量Ramsey理论研究在更复杂的度量空间中寻找大型良结构子集的问题。对于有限度量空间,该问题首先由Bourgain、Figiel和Milman \cite{bfm}研究,并由Bartal等人 \cite{BLMN03}进一步深入。本文通过一种新颖的\emph{度量Ramsey分解}概念为该问题提供确定性构造。该方法产生了若干更多应用,反映了度量嵌入理论中的一些基本结果。这些应用包括度量Ramsey理论中的各类结果,含首个给出具有紧界Ramsey定理的确定性构造,以及更强的定理与性质,隐含了相应的距离预言机应用。此外,该分解为有限度量空间到欧氏空间提供了首个确定性Bourgain型嵌入,以及到超度量的最优多重嵌入,从而改进了其在近似与在线算法中的应用。此处给出的分解、技术及其推论已在度量嵌入领域近期研究中被用于各类应用。
引用
@article{arxiv.2104.03484,
title = {Advances in Metric Ramsey Theory and its Applications},
author = {Yair Bartal},
journal= {arXiv preprint arXiv:2104.03484},
year = {2021}
}
备注
This is paper is still in stages of preparation, this version is not intended for distribution. A preliminary version of this article was written by the author in 2006, and was presented in the 2007 ICMS Workshop on Geometry and Algorithms. The basic result on constructive metric Ramsey decomposition and metric Ramsey theorem has also appeared in the author's lectures notes