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1. Introduction. Let S be a sequence of elements from the cyclic group Z m . We say S is zsf (zero-sum free) if there does not exist an m-term subsequence of S whose sum is zero. Let g(m, k) (resp. g * (m, k)) denote the least integer such that every sequence S with at least (resp. with exactly) k distinct elements and length g(m, k) (resp. g * (m, k)) must contain an m-term subsequence whose sum is zero. By an affine transformation in Z m we mean a map of the form x → ax + b, with a, b ∈ Z m and gcd(a, m) = 1. Furthermore, let E(m, s) denote the set of all equivalence classes of zsf sequences S of length s, up to order and affine transformation, that are not a proper subsequence of another zsf sequence. Using the above notation, the renowned Erdős-Ginzburg-Ziv Theorem (The function g(m, k) was introduced in [4], where it was shown that g(m, 4) = 2m − 3 for m ≥ 4. Furthermore, based on a lower bound construction the authors conjectured the value of g(m, k) for fixed k and sufficiently large m. Concerning the upper bound, they established an upper bound for m prime modulo the affirmation of the Erdős-Heilbronn conjecture (EHC). Since then, the EHC has been affirmed [9], [2], moreover, the bound given in [4] was extended for nonprimes in [19]. As will later be seen, it is worthwhile to mention that the affirmation of the EHC has resulted in several attempted generalizations and related results [6]

As a consequence of our main result, a theorem of Schrijver and Seymour that determines the zero sum Ramsey numbers for the family of all r-hypertrees on m edges and a theorem of Bialostocki and Dierker that determines the zero sum Ramsey numbers for r-hypermatchings are combined into a single theorem. Another consequence is the determination of zero sum Ramsey numbers of multiple copies of some small graphs.

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