Leírás
A set P is sphere-Ramsey if for any r any r-coloring of a sufficiently high dimensional sphere of radius R contains a monochromatic copy of P. But wait, how much should R be in this definition? Can it depend on r? Or is it rho, the radius of the smallest sphere that contains P? This latter option would be too strong for most P, but I will show that a sphere of radius rho+epsilon suffices for any finite, solvably transitive set P. Or at least, I will sketch the proof for the case when P is an equilateral triangle. The proof uses a new, topological argument, that the chromatic number of what I call Kneser-shift graph KSh(n,k) tends to infinity as n - 3k grows. Details can be found here: https://arxiv.org/abs/2608.10865.