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Inverse results for weighted Harborth constants

Abstract:

For a finite abelian group (G, +), the Harborth constant is defined as the smallest integer ℓ such that each squarefree sequence over G of length ℓ has a subsequence of length equal to the exponent of G whose terms sum to 0. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling 0 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 Erdos-Ginzburg-Ziv constant.