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On three zero-sum Ramsey-type problems

✍ Scribed by Noga Alon; Yair Caro

Publisher
John Wiley and Sons
Year
1993
Tongue
English
Weight
821 KB
Volume
17
Category
Article
ISSN
0364-9024

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

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 function f: E(K~r~) ↦ Z~k~ there is a copy G^1^ of G in K~r~ so that Σe∈E(G^1^) f(e) = 0 (in Z~k~). We prove that for every integer k~1~ R(K~n~, Z~k~) ≤ n + O(k^3^ log k) provided n is sufficiently large as a function of k and k divides (). If, in addition, k is an odd prime‐power then R(K~n~, Z~k~) ≤ n + 2__k__ ‐ 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,Z~2~) ≤ n + 2. This estimate is sharp. © 1993 John Wiley & Sons, Inc.

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