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

The Harborth constant of a finite abelian group is the smallest integer $\mathrm{\ell }$ such that each subset of $G$ of cardinality $\mathrm{\ell }$ 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.