Ultrametric subsets with large Hausdorff dimension

It is shown that for every $\text{e}\in (0,1)$, every compact metric space $(X,d)$ has a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1\mathrm{/}\text{e})$, and $${\dim }_H(S)\ge (1-\text{e}){\dim }_H(X),$$ where ${\dim }_H(\cdot )$ denotes Hausdorff dimension. The above $O(1\mathrm{/}\text{e})$ distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.