T. Banakh, O. Verbitsky
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.
@article{d3280482-ac7e-4d13-a942-750fda038e51,
title={A Ramsey T reatment of Symmetry},
author={T. Banakh and O. Verbitsky},
year={2000},
language={en}
}TY - JOUR TI - A Ramsey T reatment of Symmetry AU - T. Banakh AU - O. Verbitsky PY - 2000 LA - en ER -
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