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

Czechoslovak Mathematical Journal (2014)

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

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topChvátal, 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

top- 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
- 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
- Chiniforooshan, E., Chvátal, V., A De Bruijn-Erdős theorem and metric spaces, Discrete Math. Theor. Comput. Sci. 13 (2011), 67-74. (2011) Zbl1283.52022MR2812604
- Bruijn, N. G. De, Erdős, P., On a combinatorial problem, Proc. Akad. Wet. Amsterdam 51 (1948), 1277-1279. (1948) Zbl0032.24405MR0028289
- 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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