The oscillation stability problem for the Urysohn sphere: A combinatorial approach

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for ${\mathrm{\ell }}_2$ in the context of the Urysohn space $\text{Ur}$. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.