Some low distortion metric Ramsey problems

In this note we consider the metric Ramsey problem for the normed spaces l(p). Namely, given some 1 less than or equal to p less than or equal to infinity and alpha greater than or equal to 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 [1] 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 not equal 2. It is shown that a phase transition does occur at alpha = 2 for every p is an element of [ 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).