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Almost-monochromatic sets and the chromatic number of the plane

Frankl, Nóra, Hubai, Tamás and Pálvölgyi, Dömötör (2020) Almost-monochromatic sets and the chromatic number of the plane. In: Cabello, Sergio and Chen, Danny Z., (eds.) 36th International Symposium on Computational Geometry, SoCG 2020. Leibniz International Proceedings in Informatics, LIPIcs. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. ISBN 9783959771436

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Identification Number: 10.4230/LIPIcs.SoCG.2020.47

Abstract

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.

Item Type: Book Section
Additional Information: © 2020 The Author(s).
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 19 Aug 2022 13:30
Last Modified: 19 Aug 2022 13:33
URI: http://eprints.lse.ac.uk/id/eprint/116046

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