A zero-sum theorem
type: Item
instance of: scholarly article
title: A zero-sum theorem
zbMATH Open document ID: 1017.05102
DOI: 10.1016/S0097-3165(02)00007-9
author: Arie Bialostocki
author: Guy Bialostocki
author: Daniel Schaal
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-03-26T00:00:00Z
review text: For a nontrivial abelian group \(G\) of order \(m\), \(G\)-colorings of the integer set \([1,n]:=\{1,2,\ldots,n\}\) associating one element of \(G\) with every element of \([1,n]\) are considered. An \(m\)-tuple \((x_1,\dots,x_m)\) of integers from \([1,n]\) is called zero-sum with respect to a given coloring if the sum of the images of these integers in \(G\) is zero. It is shown that \(n=m(m-1)+1\) is the smallest integer such that every \(G\)-coloring of \([1,n]\) contains a zero-sum solution to the inequality \(x_1+x_2+\cdots+x_{m-1}<x_m\). For cyclic abelian groups \(G\) this yields an Erdős-Ginzburg-Ziv type generalization of a known result concerning a monochromatic solution of the above inequality in a 2-coloring of the positive integers.
reviewed by: Kathrin Klamroth
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 1888443
zbMATH Keywords: zero-sum
zbMATH Keywords: Erdős-Ginzburg-Ziv
zbMATH Keywords: Ramsey
zbMATH Keywords: abelian group
MaRDI profile type: Publication
cites work: Q4284607
cites work: Q4881837
cites work: On a zero-sum generalization of a variation of Schur's equation
cites work: On a variation of Schur numbers
cites work: Zero-sum problems -- a survey
cites work: Q5841589
description: scientific article; zbMATH DE number 1888443