A Variant of Harborth Constant

TL;DR

This paper investigates the $k$-Harborth constant in finite abelian groups, determining exact values or bounds for specific groups, which advances understanding of zero-sum subset problems in additive combinatorics.

Contribution

The paper introduces a variant of the Harborth constant and provides exact values or bounds for this constant in certain finite abelian groups.

Findings

Exact values of $g^k(G)$ for some groups

Lower and upper bounds for $g^k(G)$ in others

Enhanced understanding of zero-sum subset problems

Abstract

Let $G$ be a finite additive abelian group. For given $k$ a positive integer, the $k$-Harborth constant $g^k(G)$ is defined to be the smallest positive integer $t$ such that given a set $S$ of elements of $G$ with size $t$ there exists a zero-sum subset of size $k$. We find either the exact value of $g^k(G)$, or lower and upper bounds for this constant for some groups.