A Ramsey treatment of symmetry
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 \(k+1\).
scientific article; zbMATH DE number 1526880