### Abstract

This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every ε ∈ (0, 1), every n-point metric space has a subset of size n^{1-ε} which embeds into Hubert space with distortion O(1/ε). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in [32]. Namely, we show that for every n-point metric space X, and k ≥ 1, there exists an O(k)-approximate distance oracle whose storage requirement is 0(n^{1+1/k}), and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

Original language | English (US) |
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Pages (from-to) | 253-275 |

Number of pages | 23 |

Journal | Journal of the European Mathematical Society |

Volume | 9 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2007 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Approximate distance oracle
- Metric Ramsey theorem
- Proximity data structure

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## Cite this

*Journal of the European Mathematical Society*,

*9*(2), 253-275. https://doi.org/10.4171/JEMS/79