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Zero-sum Ramsey number for non-cyclic group

Abstract:

Let G be a graph with n edges and let H be a finite abelian group such that the order of each element of H divides n. Let R(G,H) denote the minimum integer t such that for every function f: E(Kt) → H there is a copy of G in Kt with the property that (Formula Presented). We prove that for a positive integer r, if H = Zrn is the abelian group of all vectors of length r over Zn, then R(G, 2r) ≤ R(G,Zrn), (0.1) where R(G, 2r) is the smallest integer N such that for every 2r-coloring of the edges of the complete graph KN, there is a monochromatic copy of G. Moreover, we shall consider the bounds to R(G,Zrn) when G the star of n edges