摘要：In a colouring of â"^d a pair (S,sâ,) with S âS â"^d and with sâ, â^^ S is almost-monochromatic if S⧵{sâ,} is monochromatic but S is not. We consider questions about finding almost-monochromatic similar copies of pairs (S,sâ,) in colourings of â"^d, â"¤^d, and of â"S under some restrictions on the colouring. Among other results, we characterise those (S,sâ,) with S âS â"¤ for which every finite colouring of â" without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of (S,sâ,). We also show that if S âS â"¤^d and sâ, is outside of the convex hull of S⧵{sâ,}, then every finite colouring of â"^d without a monochromatic similar copy of â"¤^d contains an almost-monochromatic similar copy of (S,sâ,). Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of Ï(â"²) ⥠5.
关键词：discrete geometry; Hadwiger-Nelson problem; Euclidean Ramsey theory
