Mathematics Subject Classification: 11B30, 11B75, 94B05, 94B65, 51E22.
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Artículo
Existence conditions for k-barycentric olson constant
Condiciones de existencia para la constante de olson k-baricéntrica
1Escuela Superior Politécnica del Litoral; Departamento de Matemáticas; Facultad de Ciencias Naturales; Guayaquil; Ecuador; luzelimarchan@gmail.com
2Universidad Central de Venezuela; Departamento de Matemáticas y Laboratorio LaTecS; Centro ISYS; Facultad de Ciencias; Caracas; Venezuela; oscarordaz55@gmail.com
3Universidad de Oriente; Departamento de Matemáticas; Núcleo Sucre; Cumaná; Venezuela; jsalazar@udo.edu.ve
4Universidad de Oriente; Cumaná; Departamento de Matemáticas; Núcleo Sucre; Venezuela; feliciavillarroel@gmail.com
Let (G, +) be a finite abelian group and 3 ≤ k ≤ |G| a positive integer. The k-barycentric Olson constant denoted by BO(k, G) is defined as the smallest integer ℓ such that each set A of G with |A| = ℓ contains a subset with k elements {a1, . . . , ak} satisfying a1 + · · · + ak = kaj for some 1 ≤ j ≤ k. We establish some general conditions on G assuring the existence of BO(k, G) for each 3 ≤ k ≤ |G|. In particular, from our results we can derive the existence conditions for cyclic groups and for elementary p-groups p ≥ 3. We give a special treatment over the existence condition for the elementary 2-groups.
Keywords: finite abelian group; zero-sum problem; baricentric-sum problem; Davenport constant; k-barycentric Olson constant.
Sean (G, +) un grupo abeliano finito y 3 ≤ k ≤ |G| un entero positivo. La constante de Olson k-baricéntrica, denotada por BO(k, G), se define como el menor entero positivo ℓ tal que todo conjunto A de G con |A| = ℓ contiene un subconjunto con k elementos {a1, . . . , ak} que satisface a1+· · ·+ak = kaj para algún 1 ≤ j ≤ k. Establecemos algunas condiciones generales sobre G asegurando la existencia de BO(k, G) para cada 3 ≤ k ≤ |G|. En particular, a partir de nuestros resultados podemos determinar las condiciones de existencia para los grupos cíclicos y para los p-grupos elementales con p ≥ 3. Damos un tratamiento especial a la condición de existencia para los 2-grupos elementales.
Palabras clave: grupos abelianos finitos; problemas de suma-cero; problemas de suma baricéntricas; constante de Davenport; constante k-baricéntrica de Olson.
Mathematics Subject Classification: 11B30, 11B75, 94B05, 94B65, 51E22.
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Acknowledgements
The research of O. Ordaz is supported by the Postgrado de la Facultad de Ciencias de la U.C.V., the CDCH project, and the Banco Central de Venezuela.
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Received: July 16, 2018; Revised: June 25, 2020; Accepted: October 30, 2020