A Ramsey treatment of symmetry
type: Item
instance of: scholarly article
title: A Ramsey treatment of symmetry
zbMATH Open document ID: 0960.05101
author: R. Smith
published in: The Electronic Journal of Combinatorics
publication date: 2000-11-30T00:00:00Z
full work available at URL: https://eudml.org/doc/120922
full work available at URL: v7i1r52.html
review text: For a space \(\Omega\) endowed with symmetry, \(\text{ms}(\Omega, r)\) is defined to be the maximum \(m\) such that for any \(r\)-coloring of \(\Omega\) there is a monochromatic symmetric set of size at least \(m\). The authors study a wide range of spaces \(\Omega\), noting that \(\text{ms}(\{1,\dots, n\},r)\) and \(\text{ms}([0, 1],r)\) are closely related, finding lower and upper bounds for \(\text{ms}([0, 1],2)\) and asymptotics of \(\text{ms}([0,1],r)\) for \(r\) increasing. They also find \(\text{ms}(\Omega, r)\) exactly for figures of revolution, regular polygons, and multidimensional parallelopipeds; and they show that the minimal \(r\) such that there exists an \(r\)-coloring of the \(k\)-dimensional integer grid without infinite monochromatic symmetric subsets is \(k+1\).
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 1526880
zbMATH Keywords: coloring
zbMATH Keywords: Ramsey treatment
zbMATH Keywords: symmetry
zbMATH Keywords: monochromatic symmetric set
zbMATH Keywords: polygons
zbMATH Keywords: parallelopipeds
reviewed by: Arthur T. White
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1526880