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Journal of the London Mathematical Society 2005 71(2):289-303; doi:10.1112/S0024610704006155
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© The London Mathematical Society

On Metric Ramsey-Type Dichotomies

Yair Bartal, Nathan Linial, Manor Mendel and Assaf Naor

Institute of Computer Science, Hebrew University Jerusalem 91904, Israel yair{at}cs.huji.ac.il, nati{at}cs.huji.ac.il, mendelma{at}cs.huji.ac.il
Theory Group, Microsoft Research One Microsoft Way 113/2131, Redmond, WA 98052-6399, USA anaor{at}microsoft.com

Received 2 January 2003.

The classical Ramsey theorem states that every graph contains either a large clique or a large independent set. Here similar dichotomic phenomena are investigated in the context of finite metric spaces. Namely, statements are provided of the form ‘every finite metric space contains a large subspace that is nearly equilateral or far from being equilateral’. Two distinct interpretations are considered for being ‘far from equilateral’. Proximity among metric spaces is quantified through the metric distortion {alpha}. Tight asymptotic answers are provided for these problems. In particular, it is shown that a phase transition occurs at {alpha} = 2.


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