Inverse results for weighted Harborth constants

Abstract

For a finite abelian group [Formula: see text], the Harborth constant is defined as the smallest integer [Formula: see text] such that each squarefree sequence over [Formula: see text] of length [Formula: see text] has a subsequence of length equal to the exponent of [Formula: see text] whose terms sum to [Formula: see text]. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling [Formula: see text] is required, that is, when forming the sum one can choose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constants for certain groups, in particular, for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erdős–Ginzburg–Ziv constant.