Remarks on the plus-minus weighted Davenport constant

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsf{D}_{\pm}(G)$, is the smallest $\ell$ such that each sequence $g_1 ... g_{\ell}$ over $G$ has a weighted zero-subsum with weights +1 and -1, i.e., there is a non-empty subset $I \subset \{1,..., \ell\}$ such that $\sum_{i \in I} a_i g_i =0$ for $a_i \in \{+1,-1\}$. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of groups

Inverse results for weighted Harborth constants

For a finite abelian group $(G,+)$ the Harborth constant is defined as the smallest integer $\ell$ such that each squarefree sequence over $G$ of length $\ell$ 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 constant 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 Erd\H{o}s--Ginzburg--Ziv constant

Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory

We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group $( G , + )$, a set of weights $W \subset \mathbb{Z}$, and an integral parameter $m$, the $m$-wise Davenport constant with weights $W$ is the smallest integer $n$ such that each sequence over $G$ of length $n$ has at least $m$ disjoint zero-subsums with weights $W$. And, for an integral parameter $d$, the $d$-constrained Davenport constant with weights $W$ is the smallest $n$ such that each sequence over $G$ of length $n$ has a zero-subsum with weights $W$ of size at most $d$. First, we establish a link between these two types of constants and several basic and general results on them. Then, for elementary $p$-groups, establishing a link between our constants and the parameters of linear codes as well as the cardinality of cap sets in certain projective spaces, we obtain various explicit results on the values of these constants

A weighted generalization of two theorems of Gao

Let G be a finite abelian group and let A⊆ℤ be nonempty. Let D A(G) denote the minimal integer such that any sequence over G of length D A(G) must contain a nontrivial subsequence s 1...s r such that Σ i=1r w is ifor some w i∈A. Let E A(G) denote the minimal integer such that any sequence over G of length E A(G) must contain a subsequence of length {pipe}G{pipe}, s 1...s {pipe}G{pipe}, such that Σ {pipe}G{pipe}i=1 w is i for some w i∈A. In this paper, we show that E A(G)={pipe}G{pipe}+D A(G)-1, confirming a conjecture of Thangadurai and the expectations of Adhikari et al. The case A={1} is an older result of Gao, and our result extends much partial work done by Adhikari, Rath, Chen, David, Urroz, Xia, Yuan, Zeng, and Thangadurai. Moreover, under a suitable multiplicity restriction, we show that not only can zero be represented in this manner, but an entire nontrivial subgroup, and if this subgroup is not the full group G, we obtain structural information for the sequence generalizing another non-weighted result of Gao. Our full theorem is valid for more general n-sums with n≥{pipe}G{pipe}, in addition to the case n={pipe}G{pipe}. © 2012 Springer Science+Business Media, LLC

On the Harborth constant of C3⊕C3nC_3 \oplus C_{3n}

For a finite abelian group $(G,+, 0)$ the Harborth constant \g(G) is the smallest integer $k$ such that each squarefree sequence over $G$ of length $k$, equivalently each subset of $G$ of cardinality at least $k$, has a subsequence of length $\exp(G)$ whose sum is $0$. In this paper, it is established that \g(G)= 3n + 3 for prime $n \neq 3$ and \g(C_3 \oplus C_{9})= 13