A de Bruijn-Erdős theorem for 1–2 metric spaces
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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.
Keywordsline in metric space De Bruijn-Erdős theorem
MSC 201005D99 51G99
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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