  The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other
things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension:
For any $0<\beta<\alpha$, any compact metric space $X$ of Hausdorff dimension
$\alpha$ contains a subset which is biLipschitz equivalent to an ultrametric
and has Hausdorff dimension at least $\beta$. In this note we present a simple
proof of the ultrametric skeleton theorem in doubling spaces using Bartal's
Ramsey decompositions [Bartal 2021]. The same general approach is also used to
answer a question of Zindulka [Zindulka 2020] about the existence of "nearly
ultrametric" subsets of compact spaces having full Hausdorff dimension.
