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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 } i=1 n is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence {b i } i=1 n of S such that 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 ,,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 i=1 n w i A i ' and |w i A i ' |=|A i ' | for all i, where w i A i ={w i a i a i A i }.

MSC:
11B75Combinatorial number theory
05D99Extremal combinatorics