Edge-Ramsey theory
Let \((K_n, E_n)\) be a sequence of graphs in Euclidean spaces \(\mathbb{R}^n\). It is said to be edge-Ramsey if, for all \(r\geq 1\), a positive integer \(n= n(K, E, r)\) exists so that if \(m\geq n\), for any \(r\)-coloring of the edges of \(\mathbb{R}^m\), there is a graph \((L, F)\) in \(\mathbb{R}^m\) which is geometrically congruent to \((K_m, E_m)\) so that all edges in \(F\) are assigned the same color. It is shown that the graphs of the \(n\)-dimensional measure polytope (the \(n\)-cube) and of the \(n\)-dimensional cross polytope are edge-Ramsey. Together with the \(n\)-dimensional simplices these are the only regular edge-Ramsey polytopes, with the possible exceptions of the hexagon and the 24-cell.
scientific article; zbMATH DE number 881219
scientific article