On Ramsey sets in spheres

We prove that for every vertex set L of a simplex of circumradius 1 in Rm, every r and every ϵ > 0, one can find N such that for any r-coloring of a sphere S of radius (1 + ϵ) in RN there is a monochromatic congruent copy of L in S. We also show that for almost all such configurations L, the “blowup by (1 + ϵ)” is really necessary; i.e., for any N, one can color the N-dimensional unit sphere by a certain fixed number of colors such that it contains no monochromatic congruent copy of L. The exceptional configurations for which the coloring does not work have a simple description.