Type: Article

On barycentric constants

Luca, Florian; Ordaz, Oscar; Varela, María Teresa;

Abstract:

Let G be an abelian group with n elements. Let S be a sequence of elements of G, where the repetition of elements is allowed. Let vertical bar S vertical bar be the cardinality, or the length of S. A sequence S subset of G with vertical bar S vertical bar >= 2 is barycentric or has a barycentric-sum if it contains one element alpha(j) such that Sigma(alpha i is an element of S) alpha(i) = vertical bar S vertical bar alpha(j). This paper is a survey on the following three barycentric constants: the k-barycentric Olson constant BO(k, G), which is the minimum positive integer t >= k >= 3 such that any subset of t elements of G contains a barycentric subset with k elements, provided such an integer exists; the k-barycentric Davenport constant BD (k, G), which is the minimum positive integer t such that any subsequence of t elements of G contains a barycentric subsequence with k terms; the barycentric Davenport constant BD (G), which is the minimum positive integer t >= 3 such that any subset of t elements of G contains a barycentric subset. New values and bounds on the above barycentric constants when G = Z(n) is the group of integers modulo n are also given.