On Ramsey sets in spheres (Q1347251)

It is shown that if \(L\) is the vertex set of a simplex of circumradius 1 in \(m\)-dimensional Euclidean space, and if a natural number \(r\) and \(\varepsilon > 0\) are given, then there is a natural number \(N\) such that for every \(r\)-coloring of a sphere of radius \(1 + \varepsilon\) in \(N\)- dimensional space there is a monochromatic congruent copy of \(L\). With \(\varepsilon = 0\) this is not true for most configurations.