Chromatic properties of the Euclidean plane

Let $G$ be the unit distance graph in the plane. A well-known problem in combinatorial geometry is that of determining the chromatic number of $G$. It is known that $4\le \chi (G)\le 7$. The upper bound of 7 is obtained using tilings of the plane. The present paper studies two problems where we seek proper colourings of $G$, adding restrictions inspired by tilings: Let $H(ϵ)$ be the graph whose vertices are the points of ${\mathbb R}^2$,withanedgebetweentwopointsiftheirdistanceliesintheinterval$[1,1+\epsilon]$\mathrm{.}Weshowthatforsmall$\epsilon$,$0<\epsilon\le \frac{3\sqrt{2}}{4}-1$,wehave$6\le \chi(H(\epsilon))\le 7$. This improves the result of Exoo and Grytczuk et al. that $5\le \chi (H(ϵ))$ for small $ϵ$. Suppose that $G$ is properly coloured, but so that two solidly coloured regions meet along a straight line in some neighbourhood. Then at least 5 colours must be used.