A 15-colouring of 3-space omitting distance one

AbstractIt is known that the number χ(R3) of colours necessary to colour each point of 3-space so that no two points lying distance 1 apart have the same colour lies between 5 and 18. All optimal colourings (which establish the upper bound for χ(Rn)) have to date been found using lattice–sublattice colouring schemes. This paper shows that in Rn such colouring schemes must use at least 2n+1−1 colours to have an excluded distance. In addition this paper constructs a 1-excluded colouring of R3 using a lattice–sublattice scheme with 15 colours—the least number of colours possible for such schemes