A De Bruijn-Erdős theorem for 1 - 2 metric spaces

Václav Chvátal

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 1, page 45-51
  • ISSN: 0011-4642

Abstract

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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 .

How to cite

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Chvátal, Václav. "A De Bruijn-Erdős theorem for $1$-$2$ metric spaces." Czechoslovak Mathematical Journal 64.1 (2014): 45-51. <http://eudml.org/doc/261986>.

@article{Chvátal2014,
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 $2$.},
author = {Chvátal, Václav},
journal = {Czechoslovak Mathematical Journal},
keywords = {line in metric space; De Bruijn-Erdős theorem; line in metric space; De Bruijn-Erdős theorem},
language = {eng},
number = {1},
pages = {45-51},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A De Bruijn-Erdős theorem for $1$-$2$ metric spaces},
url = {http://eudml.org/doc/261986},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Chvátal, Václav
TI - A De Bruijn-Erdős theorem for $1$-$2$ metric spaces
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 45
EP - 51
AB - 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 $2$.
LA - eng
KW - line in metric space; De Bruijn-Erdős theorem; line in metric space; De Bruijn-Erdős theorem
UR - http://eudml.org/doc/261986
ER -

References

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  1. Aboulker, P., Bondy, A., Chen, X., Chiniforooshan, E., Miao, P., 10.1016/j.dam.2014.02.008, Discrete Appl. Math. 171 (2014), 137-140. (2014) Zbl1288.05185MR3190588DOI10.1016/j.dam.2014.02.008
  2. Chen, X., Chvátal, V., 10.1016/j.dam.2007.05.036, Discrete Appl. Math. 156 (2008), 2101-2108. (2008) Zbl1157.05019MR2437004DOI10.1016/j.dam.2007.05.036
  3. Chiniforooshan, E., Chvátal, V., A De Bruijn-Erdős theorem and metric spaces, Discrete Math. Theor. Comput. Sci. 13 (2011), 67-74. (2011) Zbl1283.52022MR2812604
  4. Bruijn, N. G. De, Erdős, P., On a combinatorial problem, Proc. Akad. Wet. Amsterdam 51 (1948), 1277-1279. (1948) Zbl0032.24405MR0028289
  5. Erdős, P., Three point collinearity, Problem 4065, Am. Math. Mon. 50 (1943), 65; Solutions in vol. 51 (1944), 169-171. (1944) MR1525919

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