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On the Number of Monochromatic Lines in \(\pmb {\mathbb {R}}^d\)

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Abstract

Let X be a nonempty finite subset of \({\mathbb {R}}^d\) and \(X=\bigcup _{i=1}^m X_i\) a coloring with \(m<d\). In this paper we study the number of monochromatic lines generated by X. More specifically we give three results which establish that, under nontrivial assumptions, the number of monochromatic lines generated by X is \(\varTheta (|X|^2)\).

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Acknowledgements

We would like to thank the referees for their positive and insightful comments and advice to improve this paper.

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  1. CONACyT/UAZ, Zacatecas, Mexico

    Mario Huicochea

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  1. Mario Huicochea
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Correspondence to Mario Huicochea.

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Huicochea, M. On the Number of Monochromatic Lines in \(\pmb {\mathbb {R}}^d\). Discrete Comput Geom 65, 1061–1086 (2021). https://doi.org/10.1007/s00454-020-00210-2

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  • DOI: https://doi.org/10.1007/s00454-020-00210-2

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