Ramsey partitions of metric spaces

We investigate the existence of metric spaces which, for any coloring with a fixed number of colors, contain monochromatic isomorphic copies of a fixed starting space K. In the main theorem we construct such a space of size $2^{\aleph_0}$ for colorings with $\aleph_0$ colors and any metric space $K$ of size $\aleph_0$. We also give a slightly weaker theorem for countable ultrametric $K$ where, however, the resulting space has size~$\aleph_1$.