On Embeddings of Finite Subsets of ${\mathrm{\ell }}_p$

We study finite subsets of ${\mathrm{\ell }}_p$ and show that, up to nowhere dense and Haar null complement, all of them embed isometrically into any Banach space that uniformly contains the spaces ${\mathrm{\ell }}_p^n$, $n\in \mathbb{N}$.