Some exact values of the Harborth constant and its plus-minus weighted analogue

TL;DR

This paper calculates exact values of the Harborth constant and its plus-minus weighted version for specific finite abelian groups, mainly those combining cyclic groups with groups of order two, advancing understanding of these combinatorial invariants.

Contribution

It provides explicit values for these constants for certain groups, especially those that are direct sums of cyclic groups and groups of order two, and compares these with existing conjectures.

Findings

Exact values for Harborth constants for specific groups

Exact values for plus-minus weighted Harborth constants for these groups

Comparison with existing conjectures and results

Abstract

The Harborth constant of a finite abelian group is the smallest integer $\ell$ such that each subset of $G$ of cardinality $\ell$ has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this constant is defined in the same way except that instead of considering the sum of all elements of the subset one can choose to add either the element or its inverse. We determine these constants for certain groups, mainly groups that are the direct sum of a cyclic group and a group of order 2. Moreover, we contrast these results with existing results and conjectures on these problems.