Canonical theorems in geometric Ramsey theory

Abstract

In Euclidean Ramsey Theory usually we are looking for monochromatic configurations in the Euclidean space, whose points are colored with a fixed number of colors. In the canonical version, the number of colors is arbitrary, and we are looking for an `unavoidable' set of colorings of a finite configuration, that is a set of colorings with the property that one of them always appears in any coloring of the space. This set definitely includes the monochromatic and the rainbow colorings. In the present paper, we prove the following two results of this type. First, for any acute triangle $T$, and any coloring of $\mathbb{R}^3$, there is either a monochromatic or a rainbow copy of $T$. Second, for every $m$, there exists a sufficiently large $n$ such that in any coloring of $\mathbb{R}^n$, there exists either a monochromatic or a rainbow $m$-dimensional unit hypercube. In the maximum norm, $\ell_{\infty}$, we have a much stronger statement. For every finite $M$, there exits an $n$ such that in any coloring of $\mathbb{R}_\infty^n$, there is either a monochromatic or a rainbow isometric copy of $M$.