A 15-colouring of 3-space omitting distance one

Authors:
D. Coulson
Affiliations:
Mathematics and Statistics Department, The University of Melbourne, Melbourne 3052, Australia
Venue:
Discrete Mathematics
Year:
2002

Citing 2
Cited 1

Sphere-packings, lattices, and groups

Sphere-packings, lattices, and groups
An 18-colouring of 3-space omitting distance one

Discrete Mathematics

Axiom of choice and chromatic number of Rn

Journal of Combinatorial Theory Series A

Quantified Score

Hi-index	0.05

Visualization

Abstract

It 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.