Combinatorica

, Volume 26, Issue 4, pp 445–453

A Weighted Erdős-Ginzburg-Ziv Theorem

Original Paper

DOI: 10.1007/s00493-006-0025-y

Cite this article as:
Grynkiewicz, D.J. Combinatorica (2006) 26: 445. doi:10.1007/s00493-006-0025-y

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 \( W = {\left\{ {w_{i} } \right\}}^{n}_{{i = 1}} \) is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence \( {\left\{ {b_{i} } \right\}}^{n}_{{i = 1}} \)of S such that \( {\sum\nolimits_{i = 1}^n {w_{i} b_{i} = 0} } \). This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when m = n and wi = 1 for all i, and confirms a conjecture of Y. Caro. Furthermore, we in part verify a related conjecture of Y. Hamidoune, by showing that if S has an n-set partition A=A1, . . .,An such that |wiAi| = |Ai| for all i, then there exists a nontrivial subgroup H of G and an n-set partition A′ =A1, . . .,An of S such that \( H \subseteq {\sum\nolimits_{i = 1}^n {w_{i} {A}\ifmmode{'}\else$'$\fi_{i} } } \) and \( {\left| {w_{i} {A}\ifmmode{'}\else$'$\fi_{i} } \right|} = {\left| {{A}\ifmmode{'}\else$'$\fi_{i} } \right|} \) for all i, where wiAi={wiai |aiAi}.

Mathematics Subject Classification (2000):

11B7505D99

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Mathematics 253-37CaltechPasadena, CA 91125USA