163 problems

For a prime $p$ and a positive integer $\alpha$, define $s(p,\alpha)$ to be the least positive integer $k$ such that, for any integers $a_1,\ldots,a_k$ not congruent to $0$ modulo…

Davenport constant conjecture for intervals.

Let $G$ be a finite group, and let $\mathsf E(G)$ denote the smallest positive integer $\ell$ such that every sequence of length $\ell$ over $G$ contains $|G|$ terms whose product,…

Let $G$ be a finite group. The Gao constant ${\sf E}(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a product-one subse…

Bialostocki's conjecture. The sequence $A$ contains at least

Olson's conjecture. The invariant satisfies

Let $n\in\mathbb Z^+$ and let $\mathbb Z_n^2$ denote the direct square of the cyclic group $\mathbb Z/n\mathbb Z$. A sequence over $\mathbb Z_n^2$ is a finite sequence of elements…

Rank-three Davenport constant conjecture. Equality holds:

Krause–Zahlten conjecture. For every finite cyclic group $G$,

Let $C_n^r$ be the direct sum of $r$ cyclic groups of order $n$, and let $\mathsf{s}_{kn}(C_n^r)$ denote the smallest length guaranteeing a zero-sum subsequence of length $kn$. Kub…

Let $p$ be an odd prime, let $d\geq 2$, and let $\mathbb{F}_p^d$ denote the $d$-dimensional vector space over $\mathbb{F}_p$. For an additive group $G$, let $\mathcal{O}L(G)$ be th…

Let $R$ be a commutative unitary ring. Write $J(R)$ for its Jacobson radical, and let $\textsc{I}(\mathcal{S}_R)$ denote the Erdős–Burgess constant of the multiplicative semigroup…

Let $G$ be a finite Abelian group. Write $\eta(G)$ for the smallest integer $t\geqslant 1$ such that every sequence over $G$ of length at least $t$ contains a non-empty zero-sum su…

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

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. The Gao–Zhuang conjecture. … Here ${\mathsf{E}}(G)$ is the smallest positive integer $N$ such that every sequence over $G$ of length at least $N$ has a s…

Let $C_n$ be a cyclic group of order $n$, and let $\operatorname{\mathsf{g}}(G)$ denote the smallest integer $k$ such that every squarefree sequence over a finite abelian group $G$…

Let $C_n$ be the cyclic group of order $n$, and let $s_n(C_n^2)$ denote the smallest length of a sequence over $C_n^2$ containing a zero-sum subsequence of length $n$. Kemnitz's co…

Let $G$ be a finite abelian group, and let $A$ be a nonempty subset of $[1,\operatorname{exp}(G)-1]$. Define $\mathsf D_A(G)$ as the least integer $n$ such that every sequence over…

Let $G$ be a finite abelian group, let $|G|$ denote its order, and let $Ol(G)$ be the least integer $t$ such that every subset of $G$ with cardinality $t$ contains a nonempty subse…

Let $G$ be a finite abelian group of order $n$, and let $D(G)$ denote its Davenport constant. Theorem 1 gives an upper bound for positive integers $k$ in its stated range. The uppe…

Let $G$ be a finite abelian group with cyclic decomposition … where $d_i$ divides $d_{i+1}$. Define … The Olson–Baayen conjecture. The equality … holds for every finite abelian gro…

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…

Finite-abelian-group extension. The desired conclusion should still hold for every finite abelian group $G$ and every such $D(G)$-cover $A$. For the source result, the correspondin…

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…