Low dimensional embeddings of ultrametrics

In this note we show that every n-point ultrametric embeds with constant distortion in ℓ_p^O(logn) 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 ℓ_p ^O(k2^logn). These facts have implications to embeddings of finite metric spaces in low dimensional ℓ_p spaces in the context of metric Ramsey-type theorems.