Ramsey partitions of metric spaces

Abstract

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~\(\aleph_1\).