# A Weighted Erdős-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 $$W = {\left\{ {w_{i} } \right\}}^{n}_{{i = 1}}$$ is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence $${\left\{ {b_{i} } \right\}}^{n}_{{i = 1}}$$of S such that $${\sum\nolimits_{i = 1}^n {w_{i} b_{i} = 0} }$$. This extends the Erdős–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=A1, . . .,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 $$H \subseteq {\sum\nolimits_{i = 1}^n {w_{i} {A}\ifmmode{'}\else'\fi_{i} } }$$ and $${\left| {w_{i} {A}\ifmmode{'}\else'\fi_{i} } \right|} = {\left| {{A}\ifmmode{'}\else'\fi_{i} } \right|}$$ for all i, where w i A i ={w i a i |a i A i }.

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Correspondence to David J. Grynkiewicz.

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Grynkiewicz, D.J. A Weighted Erdős-Ginzburg-Ziv Theorem. Combinatorica 26, 445–453 (2006). https://doi.org/10.1007/s00493-006-0025-y

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• 11B75
• 05D99