Ramsey partitions of metric spaces

TL;DR

This paper constructs large metric spaces that, under any coloring with countably many colors, contain monochromatic isomorphic copies of any given countable metric space, extending Ramsey theory into metric space structures.

Contribution

It introduces the existence of large metric spaces with universal monochromatic copies for any countable metric space under countable colorings.

Findings

Constructed a space of size continuum for countably many colors

Proved a similar result for ultrametric spaces with size 

Extended Ramsey-type results to metric space settings

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~\(\aleph_1\).