A de Bruijn and Erdös property in quasi-metric spaces with four points
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Fecha
2023
Autores
Profesor/a Guía
Facultad/escuela
Idioma
en
Título de la revista
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Título del volumen
Editor
Elsevier B.V.
Nombre de Curso
Licencia CC
CC BY-NC-ND 4.0 ATTRIBUTION-NONCOMMERCIAL-NODERIVATIVES 4.0 INTERNATIONAL Deed
Licencia CC
https://creativecommons.org/licenses/by-nc-nd/4.0/
Resumen
It is a classic result that a set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chvátal conjectured in 2008 that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. One way to study lines in an space is though its betweenness. Given a quasi-metric space (V,ρ), its induced quasi-metric be-tweenness is the set of triples (x, y, z) μ V3such that ρ(x, z)=ρ(x, y) +ρ(y, z). In this work, we prove the existence of a quasi-metric space on four points a, b, c and d whose quasi-metric betweenness is = {(c, a, b), (a, b, c), (d, b, a), (b, a, d)}. This space has only three lines, none of which has four points. Moreover, we show that the betweenness of any quasi-metric space on four points with this property is isomorphic to B. Since B is not metric, we conclude that Chen and Chvatal's conjecture is valid for any metric space on four points. © 2023 Elsevier B.V.. All rights reserved.
Notas
Indexación: Scopus.
Palabras clave
Betweenness , Lines , Quasi-Metric Spaces
Citación
Procedia Computer Science. Volume 223, Pages 308 - 315. 2023. 12th Latin-American Algorithms, Graphs and Optimization Symposium, LAGOS 2023. Huatulco. 18 September 2023through 22 September 2023. Code 193000
DOI
10.1016/j.procs.2023.08.242