Geodesic
A De Bruijn-Erdős theorem for $1$-$2$ metric spaces
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 45-51
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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 $2$.
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 $2$.
DOI : 10.1007/s10587-014-0081-1
Classification : 05D99, 51G99
Keywords: line in metric space; De Bruijn-Erd\H os theorem 
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Chvátal, Vašek. A De Bruijn-Erdős theorem for $1$-$2$ metric spaces. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 45-51. doi: 10.1007/s10587-014-0081-1

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