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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 $W={\left\{{w}_{i}\right\}}_{i=1}^{n}$ is a sequence of integers whose sum is zero modulo $k$, then there exists a rearranged subsequence ${\left\{{b}_{i}\right\}}_{i=1}^{n}$ of S such that ${\sum }_{i=1}^{n}{w}_{i}{b}_{i}=0$. 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 [Discrete Math. 152, No. 1–3, 93–113 (1996; Zbl 0856.05068)]. Furthermore, we in part verify a related conjecture of 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 $A={A}_{1},\cdots ,{A}_{n}$ 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 ${A}^{\text{'}}={A}^{\text{'}}1,\cdots ,{A}_{n}^{\text{'}}$ of $S$ such that $H\subseteq {\sum }_{i=1}^{n}{w}_{i}{A}_{i}^{\text{'}}$ and $|{w}_{i}{A}_{i}^{\text{'}}|=|{A}_{i}^{\text{'}}|$ for all $i$, where ${w}_{i}{A}_{i}=\left\{{w}_{i}{a}_{i}\mid {a}_{i}\in {A}_{i}\right\}$.

##### MSC:
 11B75 Combinatorial number theory 05D99 Extremal combinatorics
##### Keywords:
Erdős-Ginzburg-Ziv theorem