Abstract
A well-known theorem in plane geometry states that any set of n non-collinear points in the plane determines at least n lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of betweenness.
In this paper, we prove that in the plane with the \(L_1\) (also called Manhattan) metric, a non-collinear set induces at least \(\lceil n/2\rceil \) lines. This is an improvement of the previous lower bound of n/37, with substantially different proof.
Supported by grant 19-04113 of the Czech Science Fountain (GACR) and by Charles University project UNCE/SCI/004.
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References
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Acknowledgment
I would like to thank Vašek Chvátal for his many insights into the topic of lines in metric spaces, as well as useful remarks concerning the full version of this paper.
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Kantor, I. (2021). Lines in the Manhattan Plane. In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics(), vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83823-2_133
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DOI: https://doi.org/10.1007/978-3-030-83823-2_133
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