An 18-colouring of 3-space omitting distance one (Q1363671)

The chromatic number of \(\mathbb{R}^n\) is the smallest number of colors needed to color the points of the space such that no two points unit distance apart obtain the same color. The best lower bound for the chromatic number of \(\mathbb{R}^3\) has been 5, the best upper bound has been 21. The paper improves the upper bound to 18. First a coloring principle is introduced based on sublattices of a lattice; then as illustration for the principle a 21-coloration of \(\mathbb{R}^3\) is given, where no two points whose distance is in (1, 1.08) obtain the same color; finally an 18-coloration of \(\mathbb{R}^3\) is given, where no two points whose distance is in (1, 1.07) obtain the same color. It may be worth mentioning that in [\textit{L. A. Székely} and \textit{N. C. Wormald}, Discrete Math. 75, No. 1-3, 343-372 (1989; Zbl 0683.05021)] where we stated without proof that \(\mathbb{R}^3\) is 21-colorable, we had a different coloration in mind. Consider a periodic 7-coloration of the plane based on a hexagonal lattice where the hexagons have diameter one, see \textit{H. Hadwiger} [Portugaliae Math. 4, 140-144 (1944; Zbl 0060.40610)]. In this plane no two points of the same color have their distance in \((1,\sqrt 7/2)\). Make of this plane a layer \(\sqrt 7/6\) thick, color the hexagonal slabs like the hexagons were colored with colors 1 through 7, repeat the same for the next layer with colors 8 through 14, and for a third layer with colors 15 through 21. Then repeat this coloration periodically. No two points whose distance is in the interval \((\sqrt {43}/6\), \(\sqrt 7/2)\) obtain the same color, and by rescaling we obtain a longer interval, (1, 1.21), which is free from distances of points of the same color.