ON 3 ZERO-SUM RAMSEY-TYPE PROBLEMS

For a graph G whose number of edges is divisible by k, let R(G, Z(k)) denote the minimum integer r such that for every f unction f: E(K(r)) bar arrow pointing right Z(k) there is a copy G' of G in K(r) so that SIGMA(e is-an-element-of E(G') f(e) = 0 (in Z(k)). We prove that for e very integer k, R(K(n), Z(k)) less-than-or-equal-to n + O(k3 log k) pr ovided n is sufficiently large as a f unction of k and k divides (2n). If, in addition, k is an odd prime-power then R(K(n), Z(k)) less-than -or-equal-to n + 2k - 2 and this is tight if k is a prime that divides n. A related result is obtained for hypergraphs. It is further shown that for every graph G on n vertices with an even number of edges R(G, Z2) less-than-or-equal-to n + 2. This estimate is sharp.