On Modified Erdős-Ginzburg-Ziv constants of finite abelian groups

Abstract

Let $ G $ be a finite abelian group with exponent $ \exp(G) $ and $ S $ be a sequence with elements of $ G $. We say $ S $ is a zero-sum sequence if the sum of the elements in $ S $ is the zero element of $ G $. For a positive integer $ t $, let $ \mathtt{s}_{t\exp(G)}(G) $ (respectively, $ \mathtt{s}'_{t\exp(G)}(G) $) denote the smallest integer $ \ell $ such that every sequence (respectively, zero-sum sequence) $ S $ over $ G $ with $ |S|\geq \ell $ contains a zero-sum subsequence of length $ t\exp(G) $. The invariant $ \mathtt{s}_{t\exp(G)}(G) $ (respectively, $ \mathtt{s}'_{t\exp(G)}(G) $) is called the Generalized Erdős-Ginzburg-Ziv constant (respectively, Modified Erdős-Ginzburg-Ziv constant) of $ G $. In this paper, we discuss the relationship between Generalized Erdős-Ginzburg-Ziv constant and Modified Erdős-Ginzburg-Ziv constant, and determine $ \mathtt{s}'_{t\exp(G)}(G) $ for some finite abelian groups.

Keywords

• zero-sum sequence
• erdős-ginzburg-ziv theorem
• generalized erdős-ginzburg-ziv constant
• modified erdős-ginzburg-ziv constant