Almost-monochromatic sets and the chromatic number of the plane

In a colouring of Rd a pair (S, s0) with S ⊆ Rd and with s0 ∈ S is almost-monochromatic if S \ {s0} is monochromatic but S is not. We consider questions about finding almost-monochromatic similar copies of pairs (S, s0) in colourings of Rd, Zd, and of Q under some restrictions on the colouring. Among other results, we characterise those (S, s0) with S ⊆ Z for which every finite colouring of R without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of (S, s0). We also show that if S ⊆ Zd and s0 is outside of the convex hull of S \ {s0}, then every finite colouring of Rd without a monochromatic similar copy of Zd contains an almost-monochromatic similar copy of (S, s0). Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of χ(R2) ≥ 5.