Almost-Monochromatic Sets and the Chromatic Number of the Plane

Frankl, Nóra and Hubai, Tamás and Pálvölgyi, Dömötör (2023) Almost-Monochromatic Sets and the Chromatic Number of the Plane. DISCRETE AND COMPUTATIONAL GEOMETRY. ISSN 0179-5376 (In Press)

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Abstract

In a colouring of Rd a pair (S,s0) with S ⊆ Rd and with s0 ∈ S is almostmonochromatic 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.

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