A theorem of Hrushovski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph R, the universal Urysohn metric space U, and other related objects. We propose a new proof of the result and show how it can be used to average out uniform and coarse embeddings of U (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides linearize the action of various isometry groups. As an application of this technique, we show that U admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.