We will present and discuss the known examples approximate Ramsey properties of normed spaces. In some cases, this can be seen as a multidimensional Borsuk-Ulam Theorem, and as a reformulation of the extreme amenability of the automorphism group of an appropriate space (a Fraisse limit). Concerning the proofs, we will sketch them for the 3 main families of spaces: $\{\ell_p^n\}_$ for $p=1,2,\infty$.