A tight threshold for metric Ramsey phenomena

In this paper, we examine the metric Ramsey problem for the normed spaces <i>l<inf>p</inf></i>: given some 1 ≤ <i>p</i> ≤ ∞, α ≥ 1 and an integer <i>n</i>, we ask for the largest <i>m</i> such that every <i>n</i>-point metric space contains an <i>m</i>-point subspace which embeds into <i>l<inf>p</inf></i> with distortion at most α. Bartal, Linial, Mendel and Naor show in [3] that in the case of 1 ≤ <i>p</i> ≤ 2, the dependence of <i>m</i> on α undergoes a phase transition at α = 2. The case of <i>p</i> > 2 was left as an open problem. We show that the phase transition occurs around α = 2 for all <i>p</i> ≥ 1. The basis of our result is a proof that there exist {1, 2} metrics which require distortion arbitrarily close to 2 for embedding into <i>l<inf>p</inf></i>. In order to show this, we develop new tools for analyzing embeddings of random metrics into <i>l<inf>p</inf></i>.