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Czechoslovak Mathematical Journal

, Volume 64, Issue 1, pp 45–51 | Cite as

A de Bruijn-Erdős theorem for 1–2 metric spaces

  • Vašek Chvátal
Article

Abstract

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals 1 or 2.

Keywords

line in metric space De Bruijn-Erdős theorem 

MSC 2010

05D99 51G99 

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References

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    P. Aboulker, A. Bondy, X. Chen, E. Chiniforooshan, P. Miao: Number of lines in hypergraphs. Discrete Appl. Math. 171 (2014), 137–140.MATHMathSciNetGoogle Scholar
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    X. Chen, V. Chvátal: Problems related to a De Bruijn-Erdős theorem. Discrete Appl. Math. 156 (2008), 2101–2108.MATHMathSciNetGoogle Scholar
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    E. Chiniforooshan, V. Chvátal: A De Bruijn-Erdős theorem and metric spaces. Discrete Math. Theor. Comput. Sci. 13 (2011), 67–74.MATHGoogle Scholar
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    N. G. De Bruijn, P. Erdős: On a combinatorial problem. Proc. Akad. Wet. Amsterdam 51 (1948), 1277–1279.MATHGoogle Scholar
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    P. Erdős: Three point collinearity, Problem 4065. Am. Math. Mon. 50 (1943), 65; Solutions in vol. 51 (1944), 169–171.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2014

Authors and Affiliations

  1. 1.Department of Computer Science and Software EngineeringConcordia UniversityMontréalCanada

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