On Metric Ramsey-type Dichotomies

Details On Metric Ramsey-type Dichotomies On Metric Ramsey-type Dichotomies

2004-06-18

arxiv.org/abs/math/0406374v1

J. London Math. Society 71(2): 289-303, 2005

Mathematics

Combinatorics

14 pages, 0 figures

Scientific paper

10.1112/S0024610704006155

The classical Ramsey theorem, states that every graph contains either a large clique or a large independent set. Here we investigate similar dichotomic phenomena in the context of finite metric spaces. Namely, we prove statements of the form "Every finite metric space contains a large subspace that is nearly quilateral or far from being equilateral". We consider two distinct interpretations for being "far from equilateral". Proximity among metric spaces is quantified through the metric distortion D. We provide tight asymptotic answers for these problems. In particular, we show that a phase transition occurs at D=2.

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