An extension of the Erdos-Ginzburg-Ziv Theorem to hypergraphs


An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S′, and set T, T ∩ S denotes the number of terms x of S with x ∈ T, and S denotes the length of S, and S \ S′ denotes the subsequence of S obtained by deleting all terms in S′. We first prove the following two additive number theory results. (1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A = A1,..., An, such that ∑i=1nAi ≤ ∑i=1n A1, - n + 1, then there exists a subsequence S′ of S, with length S′ ≥ max{ S - n + 1, 2n}, and with an n-set partition, A′ = S′1,..., A′n, such that ∑i=1n A′ii ≥ ∑i=1n Ai - n + 1. Furthermore, if ∥Ai - Aj∥ ≤ 1 for all i and j, or if Ai ≥ 3 for all i, then A′i ⊆ Ai. (2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a, b ∈ G such that (G \ {a, b}) ∩ S ≤ ⌊m/2⌋. If S ≥ 2m - 1, then there exists an m-term zero-sum subsequence S′ of S with {a} ∩ S′ ≥ ⌊m/2⌋ or {b} ∩ S′ ≥ ⌊m/2⌋. Let H be a connected, finite m-uniform hypergraph, and let f (H) (let fzs (H)) be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group ℤm) of the vertices of the complete m-uniform hypergraph Knm, there exists a subhypergraph K isomorphic to H such that every edge in K is monochromatic (such that for every edge e in K the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph H′ of H contains an edge with at least half of its vertices monovalent in H′, or if H consists of two intersecting edges, then fzs (H) = f (H). This extends the Erdos-Ginzburg-Ziv Theorem, which is the case when H is a single edge. © 2004 Elsevier Ltd. All rights reserved.

Publication Title

European Journal of Combinatorics