OAI.GrahamSpherical.full_main

The theorem states that a certain explicit set W of twelve points in the Euclidean plane has four properties. Let p = Σ_{m≥0} 10^{-(m+1)!}, u = (1−p²)/(1+p²) and v = 2p/(1+p²), so that (u,v) is a point on the unit circle. Take the four points (1,0), (−1,0), (0,1), (0,−1), and form three labelled copies: the first rotated by the angle with cosine u and sine v, the second rotated by the opposite angle (mapping (x,y) to (ux+vy, −vx+uy)), and the third left unrotated. W is the set of these twelve points, indexed by Fin 3 × Fin 4. The theorem asserts that (1) W has exactly 12 elements, so the points are distinct; (2) W lies on the unit circle centered at the origin; (3) for every positive dimension n there is a colouring of R^n with 50 colours such that no isometric embedding of W into R^n is monochromatic; and (4) W is not Euclidean Ramsey, meaning that it is false that for every positive number r of colours there exists a positive dimension n such that every r-colouring of R^n contains a monochromatic isometric copy of W.