Edge-Ramsey theory

Abstract

LetK be a configuration, a set of points in some finite-dimensional Euclidean space. Letn andk be positive integers. The notationR(K, n, r) is an abbreviation for the following statement: For everyr-coloring of the points of then-dimensional Euclidean space,Rn, a monochromatic configurationL which is congruent toK exists. A configurationK is Ramsey if the following holds: For every positive integerr, a positive integern=n(K, r) exists such that, for allmźn, R(K, m, r) holds. A configuration is spherical if it can be embedded in the surface of a sphere inn-space, providedn is sufficiently large. It is relatively easy to show that if a configuration is Ramsey, it must be spherical. Accordingly, a good fraction of the research efforts in Euclidean Ramsey theory is devoted to determining which spherical configurations are Ramsey. It is known that then-dimensional measure polytopes (the higher-dimensional analogs of a cube), then-dimensional simplex, and the regular polyhedra inR2 andR3 are Ramsey. Now letE denote a set of edges in a configurationK. The pair (K, E) is called an edge-configuration, andRe(K, E, n, r) is used as an abbreviation for the following statement: For anyr-coloring of the edges ofRn, there is an edge configuration (L, F) congruent to (K, E) so that all edges inF are assigned the same color. An edge-configuration isedge-Ramsey if, for allrź1, a positive integern=n(K, E, r) exists so that ifmźn, the statementRe(K, E, m, r) holds. IfK is a regular polytope, it is saidK isedge-Ramsey when the configuration determined by the set of edges of minimum length is edge-Ramsey. It is known that then-dimensional simplex is edge-Ramsey and that the nodes of any edge-Ramsey configuration can be partitioned into two spherical sets. Furthermore, the edges of any edge-Ramsey configuration must all have the same length. It is conjectured that the unit square is edge-Ramsey, and it is natural to ask the more general question: Which regular polytopes are edge-Ramsey? In this article it is shown that then-dimensional measure polytope and then-dimensional cross polytope are edge-Ramsey. It is also shown that these two infinite families and then-dimensional simplexes are the only regular edge-Ramsey polytopes, with the possible exceptions of the hexagon and the 24-cell.