99 problems

Let $G$ be a finite abelian group, let $\mathcal A(G)$ denote the set of minimal zero-sum sequences over $G$, and let ${\rm D}(G)$ denote the Davenport constant. A set…

Let $p$ be a prime, let $\mathbb{Z}_p$ be the cyclic group of order $p$, and let $\mathcal A(\mathbb{Z}_p)$ denote the set of minimal zero-sum sequences over this group. A set…

Let $n$ be a positive integer, let $T$ be a sequence of length $n$ over the cyclic group $\mathbb{Z}_n$, and call a subsequence of $T$ index one when its index is one. Lemke-Kleitm…

Let $G$ be the cyclic group of order $n$, and let $S$ be a minimal zero-sum sequence over $G$ of length $4$. The index $\operatorname{ind}(S)$ is the smallest number in … where…

Generalized Erdős–Ginzburg–Ziv conjecture. For every finite group $G$,

Let $p$ be a prime, and let $s_1(Z_p^3)$ denote the least integer $\ell$ such that every sequence of length $\ell$ in the finite abelian group $Z_p^3$ contains a zero-sum subsequen…

Let $n>1$ be a positive integer, and let $p$ be the smallest prime divisor of $n$. For positive integers $n$ and $k$, let … where $S$ ranges over sequences in $\mathbb{Z}_n$ of len…

Let $n>1$ and $r>0$ be integers. Define $s_r(n)$ as the least positive integer $k$ such that every sequence of $k$ integer vectors in the relevant rank-$r$ setting contains a selec…

Let $n>1$ and $r>0$ be integers. Define $w_r(n)$ as the least positive integer $k$ such that, for any $k$ integer vectors ${\bf a}_i=(a_{i1},\ldots,a_{ir})$ not congruent to…

Relative Davenport equality conjecture. One has

Davenport constant conjecture for intervals.

Gao's conjecture. For every finite abelian group $G$, one has

The Index Conjecture à la Shen et al. For $\gcd(n,6)=1$, let $S=(a_1)(a_2)(a_3)(a_4)$ be a minimal zero-sum sequence over $G$. Suppose $\gcd(n,a_i)=1$ for all $i$. Then

Eventual upper-bound conjecture. If ${\sf D}(G)\ge 4$, then there exists $k_0\in\mathbb N$ such that

Let $G$ be a finite abelian group, let $\exp(G)$ denote its exponent, and let $\mathsf{D}(G)$ denote its Davenport constant. Let $\mathsf{s}_{k\exp(G)}(G)$ be the smallest length f…

Let $G$ be a finite abelian group with rank $r(G)\geq 2$ and $\mathsf{D}(G)=\mathsf{D}^*(G)$. Let $k$ be a positive integer with $\mathsf{D}(G)-k\in[\exp(G),\mathsf{D}(G)]$. Let…

Let $G$ be a finite abelian group with rank $r(G)\geq 2$ and $\mathsf{D}(G)=\mathsf{D}^*(G)$, and let $\exp(G)$ denote its exponent. Let $\mathsf{s}_{\leq m}(G)$ be the smallest le…

Let $p$ be a prime and let $G$ be a finite $p$-group. The Loewy length $\mathsf{L}(G)$ is the nilpotency index of the Jacobson radical of the group algebra $\mathbb{F}_p[G]$, and…

Let $G$ be a finite group. Write $|G|$ for its order and let $\mathsf{E}(G)$ denote its Gao constant, namely the least integer such that every sequence over $G$ of that length cont…

For a finite group $G$, let $\mathsf{D}(G)$ be the least positive integer such that every sequence of that length over $G$ has a nonempty product-one subsequence, and let…

Consecutive Davenport constant conjecture. For every finite group $G$, it holds

Let $G$ be a finite abelian group, let $\beta=\exp(G)$ denote its exponent, and let $k\in\mathbb N$. The invariant $\mathsf{s}_{k\beta}(G)$ is the smallest length guaranteeing a ze…

Let $C_n$ be the cyclic group of order $n$, and let … be a metacyclic group. For triples $(m,n,s)\in\mathbb N\times\mathbb N\times\mathbb Z$ with multiplicative order…

Let $n\geq 2$, let $G=C_n \oplus C_n$, let $k\in [0,n-1]$, and let $S$ be a sequence of terms from $G$ with length $|S|=\operatorname{D}(G)+k-1=2n-2+k$ having no nonempty zero-sum…

Let $q$ be a positive integer and let $t$ be a parameter for which the generalized Erdős–Ginzburg–Ziv constant $4\mathsf{EGZ}({t},{q},{q})4$ and the generalized Davenport constant…