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Summary: On three zero-sum Ramsey-type problems
Noga Alon
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
and
Yair Caro
Department of Mathematics
School of Education, Haifa University- Oranim
Tivon, 36-910, Israel
Abstract
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) Zk there is a copy G of G in Kr so that
eE(G ) f(e) = 0 (in Zk). We prove that for every integer k, R(Kn, Zk) n + O(k3
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.
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