Séminaire CAESAR
de combinatoire additive




Séance du 31 mars 2011
(10 heures 30, Jussieu, salle xx):

Oscar ORDAZ (Universidad Central de Venezuela)

Barycentric Davenport constants and Ramsey Barycentric constants


Let (G,+,0) be a finite abelian group. Let S be a sequence of elements of G, where the repetition of elements is allowed, i.e, S is an element of the free abelian monoid F(G) with basis G. Let |S| be the cardinality, or the length of S. A sequence S in F(G) with |S| >= 2 is barycentric or has a barycentric-sum if it contains one element x such that the sum of the elements a in S is equal to |S| times x. We investigate the following 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 for some special groups G are given, in particular when G=Z/nZ is the group of integers modulo n. The barycentric Ramsey number BR(H,G) is the minimum positive integer r such that any coloring of the edges of the complete graph K_r by elements of G contains a subgraph H whose assigned edge colors constitute a barycentric sequence, i.e., there exists one edge whose color is the ``average'' of the colors of all edges in H. When the number of edges e(H)=0 mod exp(G), BR(H,G) are the well known zero-sum Ramsey numbers R(H,G). In this work, these Ramsey numbers are determined for some graphs, in particular, for graphs with four and five edges without isolated vertices using G=Z/nZ, where $2 <= n <= 4, and for some graphs H with e(H)=0 mod 2 using G=(Z/2Z)^s.



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