Inverse results for weighted Harborth constants

Abstract

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 chose 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.

Harborth constants for certain classes of metacyclic groups

“…Much less is known about the Harborth constant than about the EGZ constant despite their similarities. Marchan, Ordaz, Ramos, and Schmid [13], [14] computed the Harborth constants for some classes of abelian groups, in particular, g(C 2 ⊕ C 2n ) = 2n + 3, n odd 2n + 2, n even for all n ≥ 1. They also consider the plus-minus weighted analogue of the Harborth constant, in which one may add either an element or its inverse in the sums of length exp(G).…”

“…We say that S ⊆ G of size exp(G) ≤ |S| < g(G) passes if it does not fail. (See [14] for a discussion of this concept for G = C 2 ⊕C 2n .) In this section, we characterize all failing subsets of H n,m .…”

Additive Combinatorics: A Menu of Research Problems

“…Furthermore, in [150] the same authors determined all subsets A of size 2k + 1 in Z 2 × Z 2k for which 0 ∈ (2k)±A.…”

Weighted EGZ-constant for p-groups of rank 2

“…We list here some of these contributions. (i) s A (Z r 2 ) = 2 r + 1 (see [11]); (ii) s A (Z 3 ) = 4 (see [3]), s A (Z 2 3 ) = 5 (see [4]), s A (Z 3 3 ) = 9 (see [9,13]), s A (Z 4 3 ) = 21, s A (Z 5 3 ) = 41 and s A (Z 6 3 ) = 113 (see [13]); (iii) s A (Z 4 ) = 6 and s A (Z 6 ) = 8 (see [3]); s A (Z 2 4 ) = 8 (see [1]); (iv) s A (Z 2 ⊕Z 4 ) = 7 (see [14,15]), s A (Z 2 2 ⊕Z 4 ) = 8 (see [15]) and s A (Z 2 ⊕Z 6 ) = 9 (see [14]); The following results relate these two constants.…”

“…(ii) s A (Z 3 ) = η A (Z 3 ) + 3 − 1 (see [3]), s A (Z 2 3 ) = η A (Z 2 3 ) + 3 − 1 (see [4,15]); (iii) s A (Z 4 ) = η A (Z 4 ) + 4 − 1 and s A (Z 6 ) = η A (Z 6 ) + 6 − 1 (see [3]); (iv) s A (Z 2 4 ) = η A (Z 2 4 ) + 4 − 1 (see [1,15]); (v) s A (Z 2 ⊕ Z 4 ) = η A (Z 2 ⊕ Z 4 ) + 4 − 1 (see [14,15]), s A (Z 2 2 ⊕ Z 4 ) = η A (Z 2 2 ⊕ Z 4 ) + 4 − 1 (see [15]) and s A (Z 2 ⊕ Z 6 ) = η A (Z 2 ⊕ Z 6 ) + 6 − 1 (see [14]); (vi) s A (Z p s ⊕ Z p r ) = η A (Z p s ⊕ Z p r ) + p r − 1, where p is an odd prime number and s ∈ {1, 2} (see [5,6]);…”