Advances in Metric Ramsey Theory and its Applications

TL;DR

This paper introduces a deterministic approach to metric Ramsey theory using a novel metric Ramsey decomposition, leading to new bounds, embeddings, and applications in metric space analysis and algorithms.

Contribution

It presents the first deterministic constructions for metric Ramsey theorems, including embeddings into Euclidean space and ultrametrics, advancing theoretical and practical applications.

Findings

Deterministic constructions for metric Ramsey theorems with tight bounds

First deterministic Bourgain-type embedding into Euclidean space

Optimal multi-embedding into ultrametrics for improved algorithms

Abstract

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.