A weighted Erdos-Ginzburg-Ziv theorem


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 so that they can be considered as sets. If S is a sequence of m+n-1 elements from a finite abelian group G of order m and exponent k and if W = {wi}i=1n is a sequence of integers whose sum is zero modulo k then there exists a rearranged subsequence {bi} i=1n of S such that ∑i=1n w ibi = 0. This extends the Erdos-Ginzburg-Ziv Theorem which is the case when m = n and w i = 1 for all i and confirms a conjecture of Y. Caro. Furthermore we in part verify a related conjecture of Y. Hamidoune by showing that if S has an n-set partition A=A 1 . . .A n such that |w i A i | = |A i | for all i then there exists a nontrivial subgroup H of G and an n-set partition A =A1 . . .A n of S such that H ⊆i=1n wiA′i and |wiA′ i| = |A′i| for all i where w i A i ={w i a i |a i A i }. © Springer-Verlag Berlin Heidelberg 2006.

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