Almost-Monochromatic Sets and the Chromatic Number of the Plane

Abstract In a colouring of $${\mathbb {R}}^d$$ R d pair $$(S,s_0)$$ ( S , s 0 ) with $$S\subseteq {\mathbb ⊆ and $$s_0\in S$$ ∈ is almost-monochromatic if $$S\setminus \{s_0\}$$ \ { } monochromatic but S not. We consider questions about finding similar copies pairs in colourings , {Z}}^d$$ Z {Q}}$$ Q under some restrictions on the colouring. Among other results, we characterise those {Z}}$$ for which every finite {R}}$$ without an infinite arithmetic progression contains copy . also show that $$s_0$$ outside convex hull then Further, propose approach based sets might lead to human-verifiable proof $$\chi ({{\mathbb {R}}}^2)\ge 5$$ χ 2 ≥ 5