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Almost-Monochromatic Sets and the Chromatic Number of the Plane

Authors Nóra Frankl, Tamás Hubai, Dömötör Pálvölgyi

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ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Computational geometry
Keywords
  • discrete geometry
  • Hadwiger-Nelson problem
  • Euclidean Ramsey theory

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Abstract

In a colouring of ℝ^d a pair (S,s₀) with S ⊆ ℝ^d 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 ℝ^d, ℤ^d, 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 ⊆ ℤ^d and s₀ is outside of the convex hull of S⧵{s₀}, then every finite colouring of ℝ^d without a monochromatic similar copy of ℤ^d 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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Nóra Frankl, Tamás Hubai, and Dömötör Pálvölgyi. Almost-Monochromatic Sets and the Chromatic Number of the Plane. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.47

Author Details

Nóra Frankl
  • Department of Mathematics, London School of Economics and Political Science, UK
  • Laboratory of Combinatorial and Geometric Structures at MIPT, Moscow, Russia
Tamás Hubai
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE), Budapest, Hungary
Dömötör Pálvölgyi
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE), Budapest, Hungary

Funding

  • Frankl, Nóra: Research was part of project 2018-2.1.1-UK_GYAK-2018-00024 of the summer internship for Hungarian students studying in the UK, supported by the National Research, Development and Innovation Office. She also acknowledges the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926. The research was also partially supported by the National Research, Development, and Innovation Office, NKFIH Grant K119670.
  • Hubai, Tamás: Research supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under grant number LP2017-19/2017.
  • Pálvölgyi, Dömötör: Research supported by the Lendület program of the Hungarian Academy of Sciences (MTA), under grant number LP2017-19/2017.

Acknowledgements

We thank Konrad Swanepoel and the anonymous referees for helpful suggestions on improving the presentation of the paper. We also thank the participants of the Polymath16 project for discussions and consent to publish these related results that were obtained "offline" and separately from the main project.

References

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