## 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 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 $$A=A_1,\dots,A_n$$ such that $$| w_iA_i| = | A_i|$$ 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 $$| w_i A'_i| =| A'_i|$$ for all $$i$$, where $$w_i A_i=\{w_i a_i \mid a_i \in A_i\}$$.

### MSC:

 11B75 Other combinatorial number theory 05D99 Extremal combinatorics

### Keywords:

Erdős-Ginzburg-Ziv theorem

### Citations:

Zbl 0856.05068; Zbl 0872.11016
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