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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 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
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