A Ramsey T reatment of Symmetry

Given a space Ω endowed with symmetry, we define ms(Ω, r) to be the maximum of m such that for any r-coloring of Ω there exists a monochromatic symmetric set of size at least m. We consider a wide range of spaces Ω including the discrete and continuous segments {1,...,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,...,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(Ω, 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.