A Ramsey Treatment of Symmetry

Abstract

Given a space $\Omega$ endowed with symmetry, we define $ms(\Omega, r)$ to be the maximum of $m$ such that for any $r$-coloring of $\Omega$ there exists a monochromatic symmetric set of size at least $m$. We consider a wide range of spaces $\Omega$ including the discrete and continuous segments $\{1, \ldots, n\}$ and $[0,1]$ with central symmetry, geometric figures with the usual symmetries of Euclidean space, and Abelian groups with a natural notion of central symmetry. We observe that $ms(\{1, \ldots, n\}, r)$ and $ms([0,1], r)$ are closely related, prove lower and upper bounds for $ms([0,1], 2)$, and find asymptotics of $ms([0,1], r)$ for $r$ increasing. The exact value of $ms(\Omega, r)$ is determined for figures of revolution, regular polygons, and multi-dimensional parallelopipeds. We also discuss problems of a slightly different flavor and, in particular, prove that the minimal $r$ such that there exists an $r$-coloring of the $k$-dimensional integer grid without infinite monochromatic symmetric subsets is $k+1$.