# A Weighted Erdős-Ginzburg-Ziv Theorem

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DOI: 10.1007/s00493-006-0025-y

- Cite this article as:
- Grynkiewicz, D.J. Combinatorica (2006) 26: 445. doi:10.1007/s00493-006-0025-y

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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*=*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*′ =*A*′_{1}, . . .,*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}}.