Low dimensional embeddings of ultrametrics

In this note we show that every n-point ultrametric embeds with constant distortion in ℓpO(log n) for every ∞≥p≥1. More precisely, we consider a special type of ultrametric with hierarchical structure called a k-hierarchically well-separated tree (k-HST). We show that any k-HST can be embedded with distortion at most 1+O(1/k) in ℓpO(k2log n). These facts have implications to embeddings of finite metric spaces in low dimensional ℓp spaces in the context of metric Ramsey-type theorems.