1. Introduction. Let F (G) denote the free abelian monoid over the set G with monoid operation written multiplicatively and given by concatenation, i.e., F (G) consists of all finite sequences over G modulo the equivalence relation allowing terms to be permuted. Despite possible confusion, the elements of F (G) will be referred to simply as sequences, and if indeed order or being infinite are needed in a sequence, it will be explicitly stated when the sequence is first introduced. Now let G be an abelian group of order m ≥ 2. The Erdős-GinzburgZiv theorem states that every sequence in G of length 2m − 1 contains an m-term subsequence with zero sum [5]. There have been many related inverse theorems describing the structure of the sequences S in G with length |S| = m + k, 1 ≤ k ≤ m − 2, not having any m-term subsequence with zero sum. For cyclic groups of order m, the structure of S has been described by several authors: when k = m − 2, by Yuster and Peterson in [15], and by Bialostocki and Dierker in [1]; when k = m − 3, by Flores and Ordaz in [7]; when m − ⌊m/4⌋ − 2 ≤ k ≤ m − 2, by Bialostocki, Dierker, Grynkiewicz, and Lotspeich in [2] (using a related result of Gao from [8]); and when k ≥ ⌈(m − 1)/2⌉, by Chen and Savchev in [3].
<span>El objetivo principal de esta investigación es el de presentar los juegos como estrategia metodológica en la enseñanza de la geometría, con el propósito de mejorar el rendimiento escolar de la geometría en séptimo grado de Educación Básica en la U.E.L.B “Ricardo Márquez Moreno”, ubicada en Santa Ana, estado Nueva Esparta, República Bolivariana de Venezuela, durante el año escolar 2008-2009. El presente estudio se enmarca en la modalidad de investigación de campo de tipo descriptivo. La población la conformaron doscientos (200) estudiantes y ocho (08) docentes del área de matemáticas. La muestra estuvo representada por 50 estudiantes integrantes de dos (2) secciones. Los instrumentos utilizados para recabar la información fueron dos cuestionarios, uno aplicado a los docentes y el otro a los estudiantes. El análisis de los resultados indicó que los docentes utilizan estrategias tradicionales para la enseñanza de la geometría como, por ejemplo, la exposición y muy pocas veces ponen en práctica la estrategia de los juegos. Además se determinó que los estudiantes necesitan motivación e integración hacia la geometría mediante estrategias motivadoras y agradables como los juegos didácticos, por lo cual se sugiere el uso de estas estrategias para mejorar el rendimiento y la calidad educativa.</span>
Let G be an abelian finite group and H be a graph. A sequence in G, with length al least two, is barycentric if it contains an "average" element of its terms. Within the context of these sequences, one defines the barycentric Ramsey number, denoted by BR(H, G), as the smallest positive integer t such that any coloration of the edges of the complete graph Kt with elements of G produces a barycentric copy of the graph H. In this work we present a method based on the combinatorial theory and on the definition of barycentric Ramsey for calculating exact values of the above metioned constant, for some small graphs where the order is less than or equal to 8. We will exemplify the case where H is the star graph K 1,k , and where G is the cyclical group Zn, with 3 ≤ n ≤ 11 and 3 ≤ k ≤ n.
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