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on Chromatic number of the plane ($C_{27a}$)

The statement

$C_{27}$ is the chromatic number of the plane, usually denoted $\chi(\mathbb R^{2})$ (the Hadwiger–Nelson problem).

Equivalently, let $U_{2}$ be the unit-distance graph on $\mathbb R^{2}$ : its vertex set is $\mathbb R^{2}$ , with an edge between distinct points $x,y\in\mathbb R^{2}$ iff $\|x-y\|_{2}=1$ . Then

C_{27} \ :=\ \chi(\mathbb R^{2}) \ :=\ \chi(U_{2}),

i.e. $C_{27}$ is the smallest integer $k$ such that there exists a map $c:\mathbb R^{2}\to\{1,\dots,k\}$ satisfying $c(x)\neq c(y)$ whenever $\|x-y\|_{2}=1$ .

It is known that

5\ \le\ C_{27}\ \le\ 7.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.