Metric Ramsey theory is concerned with finding large well-structured subsets of more complex metric spaces. For finite metric spaces this problem was first studies by Bourgain, Figiel and Milman \cite{bfm}, and studied further in depth by Bartal et. al \cite{BLMN03}. In this paper we provide deterministic constructions for this problem via a novel notion of \emph{metric Ramsey decomposition}. This method yields several more applications, reflecting on some basic results in metric embedding theory. The applications include various results in metric Ramsey theory including the first deterministic construction yielding Ramsey theorems with tight bounds, a well as stronger theorems and properties, implying appropriate distance oracle applications. In addition, this decomposition provides the first deterministic Bourgain-type embedding of finite metric spaces into Euclidean space, and an optimal multi-embedding into ultrametrics, thus improving its applications in approximation and online algorithms. The decomposition presented here, the techniques and its consequences have already been used in recent research in the field of metric embedding for various applications.

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公制拉姆齐理论及其应用进展

度量Ramsey理论与寻找更复杂度量空间的大型结构良好的子集有关。对于有限度量空间，此问题首先由Bourgain，Figiel和Milman \ cite {bfm}研究，然后由Bartal等人进一步深入研究。al \ cite {BLMN03}。在本文中，我们通过\ emph {metric Ramsey分解}的新颖概念提供了针对此问题的确定性构造。这种方法产生了更多的应用，反映了度量嵌入理论的一些基本结果。这些应用包括公制Ramsey理论中的各种结果，包括第一个确定性构造产生具有紧密边界的Ramsey定理，更强的定理和性质，这意味着适当的距离预言应用。此外，这种分解为有限度量空间到欧几里得空间提供了确定性的布尔加因类型的嵌入，并为超度量提供了最佳的多重嵌入，从而改善了其在逼近和在线算法中的应用。此处介绍的分解，技术及其后果已在各种应用的度量嵌入领域中的最新研究中使用。