Almost-monochromatic sets and the chromatic number of the plane

In a colouring of ${\mathbb{R}}^d$ a pair $(S,s_0)$ with $S\subseteq $\mathbb{R}$^d$andwith$s_0\in S$is\mathit{\text{almost}}\mathit{\text{monochromatic}}if$S\setminus {s_0}$ismonochromaticbut$S$ is not. We consider questions about finding almost monochromatic similar copies of pairs $(S,s_0)$ in colourings of ${\mathbb{R}}^d$, ${\mathbb{Z}}^d$, and in $\mathbb{Q}$ under some restrictions on the colouring. Among other results, we characterise those $(S,s_0)$ with $S\subseteq $\mathbb{Z}$$forwhicheveryfinitecolouringof$$\mathbb{R}$$ without an infinite monochromatic arithmetic progression contains an almost monochromatic similar copy of $(S,s_0)$. We also show that if $S\subseteq {\mathbb{Z}}^d$ and $s_0$ is outside of the convex hull of $S\setminus \{s_0\}$, then every finite colouring of ${\mathbb{R}}^d$ without a similar monochromatic copy of ${\mathbb{Z}}^d$ contains an almost monochromatic similar copy of $(S,s_0)$. Further, we propose an approach of finding almost-monochromatic sets that might lead to a non-computer assisted proof of $\chi ({\mathbb{R}}^2)\ge 5$.