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On some low distortion metric Ramsey problems

Yair Bartal·Nathan Linial. Manor Mendel·Assaf Naor
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Abstract

In this note, we consider the metric Ramsey problem for the normed spaces l_p. Namely, given some 1<=p<=infinity and alpha>=1, and an integer n, we ask for the largest m such that every n-point metric space contains an m-point subspace which embeds into l_p with distortion at most alpha. In [arXiv:math.MG/0406353] it is shown that in the case of l_2, the dependence of $m$ on alpha undergoes a phase transition at alpha=2. Here we consider this problem for other l_p, and specifically the occurrence of a phase transition for p other than 2. It is shown that a phase transition does occur at alpha=2 for every p in the interval [1,2]. For p>2 we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every 1<p<infinity there are arbitrarily large metric spaces, no four points of which embed isometrically in l_p.

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Yair Bartal, Nathan Linial. Manor Mendel, Assaf Naor. 2004-06-17. On some low distortion metric Ramsey problems. https://doi.org/10.1007/s00454-004-1100-z

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