Almost-monochromatic sets and the chromatic number of the plane
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In a colouring of R^d a pair (S,s_0) with S⊆R^d and with s_0∈ S is almost monochromatic if S∖{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 R^d, Z^d, and in Q under some restrictions on the colouring. Among other results, we characterise those (S,s_0) with S⊆Z for which every finite colouring of R without an infinite monochromatic arithmetic progression contains an almost monochromatic similar copy of (S,s_0). We also show that if S⊆Z^d and s_0 is outside of the convex hull of S∖{s_0}, then every finite colouring of R^d without a similar monochromatic copy of 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 χ(^2)≥ 5.
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