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 2ℵ0 for colorings with ℵ colors and any metric space K of size ℵ. We also give a slightly weaker theorem for countable ultrametric K where, however, the resulting space has size ℵ₁.