On three zero-sum Ramsey-type problems

For a graph G whose number of edges is divisible by k, let R(G,Zk) denote the minimum integer r such that for every function f : E(Kr) 7→ Zk there is a copy G′ of G in Kr so that ∑ e∈E(G′) f(e) = 0 (in Zk). We prove that for every integer k, R(Kn, Zk) ≤ n + O(k log k) provided n is sufficiently large as a function of k and k divides ( n 2 ) . If, in addition, k is an odd prime-power then R(Kn, Zk) ≤ 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) ≤ n+ 2. This estimate is sharp. ∗Research supported in part by a United States Israel BSF Grant and by a Bergmann Memorial Grant