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

Sumsets, Zero-Sums and Extremal Combinatorics

This thesis develops and applies a method of tackling zero-sum additive questions, especially those related to the Erdos-Ginzburg-Ziv Theorem (EGZ), through the use of partitioning sequences into sets, i.e., set partitions. Much of the research can alternatively be found in the literature spread across nine separate articles, but is here collected into one cohesive work augmented by additional exposition. Highlights include a new combinatorial proof of Kneser's Theorem (not currently located elsewhere); a proof of Caro's conjectured weighted Erdos-Ginzburg-Ziv Theorem; a partition analog of the Cauchy-Davenport Theorem that encompasses several results of Mann, Olson, Bollobas and Leader, and Hamidoune; a refinement of EGZ showing that an essentially dichromatic sequence of 2m-1 terms from an abelian group of order m contains a mostly monochromatic m-term zero-sum subsequence; an interpretation of Kemperman's Structure Theorem (KST) for critical pairs (i.e., those finite subsets A and B of an abelian group with |A+B|<|A|+|B|) through quasi-periodic decompositions, which establishes certain canonical aspects of KST and facilitates its use in practice; a draining theorem for set partitions showing that a set partition of large cardinality sumset can have several elements removed from its terms and still have the sumset remain of large cardinality; a proof of a subsequence sum conjecture of Hamidoune; the determination of the g(m,k) function introduced by Bialostocki and Lotspeich (defined as the least n so that a sequence of terms from Z/mZ of length n with at least k distinct terms must contain an m-term zero-sum subsequence) for m large with respect to k; the determination of g(m,5) for all m, including the details to the abbreviated proof found in the literature; various zero-sum results concerning modifications to the nondecreasing diameter problem of Bialostocki, Erdos, and Lefmann; an extension of EGZ to a class of hypergraphs; and a lower bound on the number of zero-sum m-term subsequences in a sequence of n terms from an abelian group of order m that establishes Bialostocki's conjectured value for small n<(19/3)m