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A weighted Erdős-Ginzburg-Ziv theorem. (English) Zbl 1121.11018
Summary: 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 $W=\{w_i\}^n_{i=1}$ is a sequence of integers whose sum is zero modulo $k$, then there exists a rearranged subsequence $\{b_i\}^n_{i=1}$ of S such that $\sum^n_{i=1} w_i b_i=0$. 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 {\it Y. Caro} [Discrete Math. 152, No. 1--3, 93--113 (1996; Zbl 0856.05068)]. Furthermore, we in part verify a related conjecture of {\it Y. O. Hamidoune} [Discrete Math. 162, No. 1--3, 127--132 (1996; Zbl 0872.11016)] by showing that if $S$ has an $n$-set partition $A=A_1,\dots,A_n$ such that $\vert w_iA_i\vert = \vert A_i\vert $ for all $i$, then there exists a nontrivial subgroup $H$ of $G$ and an $n$-set partition $A' =A'1,\dots,A'_n$ of $S$ such that $H\subseteq \sum^n_{i=1} w_i A'_i$ and $\vert w_i A'_i\vert =\vert A'_i\vert $ for all $i$, where $w_i A_i=\{w_i a_i \mid a_i \in A_i\}$.

11B75Combinatorial number theory
05D99Extremal combinatorics
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