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 ≤ p ≤ ∞ and α ≥ 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 α. In [1] it is shown that in the case of ℓ₂, the dependence of m on α undergoes a phase transition at α = 2. Here we consider this problem for other ℓ_p, and specifically the occurrence of a phase transition for p ≠ 2. It is shown that a phase transition does occur at α = 2 for every p ∈ [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 < ∞ there are arbitrarily large metric spaces, no four points of which embed isometrically in ℓ_p.