  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.
