19 problems

Let $H$ be a subgroup of a finite group $G$, and put $n=|G|/|H|$. An $H$-sum sequence is a sequence of elements whose sum lies in $H$. For an element $g$, let $|g|$ denote its orde…

Let $G$ be a finite group, and let $(g_k)_{k=1}^n$ be a sequence of $n$ elements of $G$, where $n=|G|$. For an element $g$, let $|g|$ denote its order, and call a subsequence zero-…

Let $h\geq 3$ and let $\mathbf{b}=(b_1,\ldots,b_h)\in\mathbb{Z}^h$ satisfy $\sum_{i=1}^h b_i=0$ and $b_i\neq 0$ for all $i$. Let $A\subseteq\mathbb{N}$, with increasing elements…

Schefler56Zhao56Zhong's conjecture. The support of $A$ has full size:

Let $G$ and $G'$ be finite abelian groups. Write $\mathcal{B}(G)$ for the monoid of zero-sum sequences over $G$, $\mathcal{L}(\mathcal{B}(G))$ for its system of sets of lengths, an…

Davenport constant conjecture. For every positive integer $n$ and every positive integer $r$, one has

Let $G$ be a finite abelian group with Davenport constant $mathsf D(G)geq 4$, and let $G'$ be an abelian group such that … Here, $\mathcal B(G)$ denotes the monoid of zero-sum seq…

Let $r$ and $s$ be fixed positive integers. For a function $f:[n]\to\{-r,s\}$ with $f([n])=0$, let $M(r,s,k)$ denote the minimum threshold such that every sufficiently large $n$ fo…

Let $r$, $s$, and $k$ be positive integers such that $r+s$ divides $k$. For a function $f:[n]\to\{-r,s\}$ with $f([n])=0$, let $M(r,s,k)$ be the least positive integer such that ev…

Let $r$, $s$, and $k$ be positive integers such that $r+s$ divides $k$. For a function $f:[n]\to\{-r,s\}$, write $f([n])=\sum_{i=1}^n f(i)$, and call $k$ consecutive indices a zero…

Let $G\cong C_n^r$ with $\mathsf D(G)\geq 4$, where … for pairwise distinct primes $p_1,\ldots,p_u$ and $u,k_1,\ldots,k_u\in\mathbb N$. Let $\mathsf K(G,r-1)$ denote the cross-numb…

Let $H$ be a transfer Krull monoid over a finite abelian group $G$, and let $\mathcal B(G)$ be the monoid of zero-sum sequences over $G$. Write $\mathsf L(a)$ for the set of factor…

Let $r$, $s$ and $k$ be positive integers such that $(r+s)$ divides $k$. A function $f:[n]\to\{-r,s\}$ has total sum $f([n])=0$ when the sum of its values over $[n]$ is zero. A ZS…

Systems-of-sets-of-lengths conjecture. If $G'$ is an abelian group satisfying

Extremal equality conjecture. Cyclic groups are the only groups for which equality holds on the left-hand side, whereas, for every noncyclic group, there exists $k^*\in\mathbb N$ s…

Structural conjecture. Each minimal zero-sum sequence $U$ over $G$ with $\mathsf{k}(U)=\mathsf{K}(G)$ has the form

For a positive integer $n$, let $\mathsf{I}(n)$ denote the maximal index of a minimal zero-sum sequence over a cyclic group of order $n$. Gao's logarithmic conjecture. There is an…

Index conjecture. If $\gcd(|G|,6)=1$, then every minimal zero-sum sequence $S$ over $G$ of length $|S|=4$ has $\operatorname{ind}(S)=1$. This conjecture concerns the index invarian…

Let $p$ be a prime, and let $S$ be a minimal zero-sum sequence over $C_p\oplus C_p$ of length $2p-1$. A minimal zero-sum sequence is a zero-sum sequence with no nonempty proper zer…