  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.
