On Metric Ramsey-type Dichotomies
نویسندگان
چکیده
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 equilateral 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 α. We provide tight asymptotic answers for these problems. In particular, we show that a phase transition occurs at α = 2.
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