Almost-Monochromatic Sets and the Chromatic Number of the Plane

In a colouring of $${\mathbb {R}}^d$$ a pair $$(S,s_0)$$ with $$S\subseteq {\mathbb {R}}^d$$ and with $$s_0\in S$$ is almost-monochromatic if $$S\setminus \{s_0\}$$ is monochromatic but 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 of $${\mathbb {Q}}$$ under some restrictions on the colouring. Among other results, we characterise those $$(S,s_0)$$ with $$S\subseteq {\mathbb {Z}}$$ for which every finite colouring of $${\mathbb {R}}$$ 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 monochromatic similar copy of $${\mathbb {Z}}^d$$ contains an almost-monochromatic similar copy of $$(S,s_0)$$ . Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of $$\chi ({{\mathbb {R}}}^2)\ge 5$$ .