not bi-Lipschitzly embed into $X$. Nowadays this result can be deduced from Bourgain’s result and a result of Ostrovskii’s [@Os1 Theorem 1.2] which states that a locally finite metric space $A$ embeds bi-Lipschitzly into a Banach space $X$ if all of its finite subsets uniformly bi-Lipschitzly embed into $X$. In [@JS] Johnson and Schechtman characterized superflexivity, using the [*Diamond Graphs*]{}, $D_n$, $n\in{\mathbb{N}}$, and proved that Banach space $X$ is super reflexive if and only if the $D_n$, $n\in{\mathbb{N}}$, do not uniformly bi-Lipschitzly embed into $X$. There are several other local properties, [*i.e.,* ]{}properties of the finite dimensional subspaces of Banach spaces, for which metric characterizations were found. The following are some examples: Bourgain, Milman and Wolfson [@BMW] characterized having non trivial type using [*Hammond cubes*]{} (the sets $B_n$, together with the $\ell_1$-norm), and Mendel and Naor [@MN1; @MN2] present metric characterizations of Banach spaces with type $p$, $1<p\le 2$, and cotype $q$, $2\le q<\infty$. For a more extensive account on the Ribe program we would like refer the reader to the survey articles [@Ball; @Na] and the book [@Os2].

Instead of only asking for metric characterizations of local properties, one can also ask for metric characterizations of other properties of 