---

Statement

Theorem 5 (p. 11). "There is a 7-fold colouring of $G_{[1,1]}$ with 37 colours i.e. $\frac{37}{7}\approx5.285$."

Here $G_{[1,1]}$ is the unit distance graph of the plane and a $j$-fold colouring is as in Definition 2 (p. 4). Table 3 (p. 15) lists this as the paper's best ratio for $j\le7$, printed there as $5.26$.

Source. J. Grytczuk, K. Junosza-Szaniawski, J. Sokół, K. Węsek, Fractional and $j$-fold coloring of the plane, Discrete Comput. Geom. 55 (2016), 594-609, doi:10.1007/s00454-016-9769-3; read in arXiv:1506.01887v2 (5 October 2015), Theorem 5 and its proof on p. 11 of that version. The source card records the edition.

Read depth. Claims checked: the statement was read on the print. The proof was read for structure only; its distance claims were not checked.

Proof pointer

p. 11. Tile the plane by hexagons of side $s=\frac{1}{2\sqrt7}$. One colour class is a periodic pattern of seven-hexagon clusters, each of diameter 1 with half its border included, whose clusters are $s\sqrt{31}=\frac{\sqrt{31}}{2\sqrt7}\approx1.05$ apart (Figure 7). The pattern is shifted 37 times by $[\frac{\sqrt3}{2\sqrt7},0]$, one shift per colour, and each hexagon receives 7 of the 37 colours.

Dependencies

None.

Bears on

• Problem 508: the problem asks for the chromatic number of $G_{[1,1]}$. Theorem 5 bounds the 7-fold chromatic number of $G_{[1,1]}$ by 37, and so its fractional chromatic number by $37/7$; it gives no bound on the chromatic number itself.