Barycentric Ramsey Numbers for Small Graphs

Let G be a finite abelian group of order 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 Kr by elements of G contains a subgraph H whose assigned edge color constitutes a barycentric sequence, i.e. there exists one edge whose color is the “average” of the colors of its edges. These BR(H,G) are determined for some graphs, in particular for graphs with at most four edges without isolated vertices (i.e. small graphs) and G = Zn, 2 ≤ n ≤ 5. Elementary combinatorial arguments are used for these computations.