124 problems

Let $s\geq 3$ be an integer. An $s$-spanning set of size $2$ in a finite abelian group is a set of two elements whose signed sums of at most $s$ terms cover the group. Largest nonc…

For every nontrivial finite abelian group $G$, let $\mathsf d(G)$ denote the maximum length of a zero-sum-free sequence over $G$. Let $\nu(G)$ be the least integer $\ell$ such that…

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

Tannenbaum's zero-sum partition conjecture. For every positive integer $t$ and every integer partition $\{m_i\}_{i=1}^t$ of $m-1$, with $m_i\geq 2$ for every $i\in[1,|R|/2]$ and…

Let $k$ be a positive integer and write $S(k)=\{1,2,\dots,k\}$. Let $G$ be a finite abelian group. Say that $S(k)$ splits $G$ if there is a subset $T\subseteq G$ such that the tran…

Balister–Győri–Schelp conjecture. There exists a partition of $\mathbb F_2^k$ into 2-sets $\{w_i,z_i\}$, for $i\in\{1,\dots,2^{k-1}\}$, such that

Let $\overrightarrow{G}$ be a digraph of order $n$ whose weakly connected components all have order at least $3$, and let $\Gamma$ be an Abelian group. A $\Gamma$-irregular labelin…

Let $\Gamma$ be a finite Abelian group, and let $I(\Gamma)$ denote its set of involutions. A finite Abelian group has 4-ZSPP if every integer partition of $|\Gamma|-1$ whose parts…

A $\Gamma$-magic rectangle set $\operatorname{MRS}_{\Gamma}(a,b;c)$ on an Abelian group $\Gamma$ of order $abc$ is a collection of $c$ arrays of size $a\times b$, whose entries are…

Klein four-group phylogenetic complexity conjecture. The phylogenetic complexity of $G$ is

Krause–Zahlten conjecture. The equality

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…

Characterization conjecture. If $\mathsf D(G)\geq 4$, then $G$ and $G'$ are isomorphic.

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 $G$ be a finite Abelian group, let $V_g f$ denote the time-frequency representation appearing in the paper, let $\widehat f$ denote the Fourier transform of $f$, and write…

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…

For an integer $n\geqslant2$, let $\mathbb{Z}_{\!/2}^n$ denote the direct sum of $n$ copies of the cyclic group $\mathbb{Z}_{\!/2}$. Let $\lambda(\mathbb{Z}_{\!/2}^n)$ denote its L…

For an integer $n\geqslant2$, let $\rho(n)$ denote the smallest prime number that does not divide $n$. For the cyclic group $\mathbb{Z}_{\!/n}$, let $\lambda(\mathbb{Z}_{\!/n})$ de…

Let $n_1\geq n_2$ be positive integers, let $0<u<1$, and let $G(n_1,n_2,u)$ be the Erdős–Rényi random bipartite graph whose bipartition has vertex sets of sizes $n_1$ and $n_2$, wi…

Higher-rank square-modulus conjecture. For every odd prime $p$ and every $r\geq 1$,

Odd square-plane conjecture. For every odd prime $p$,

Relative Davenport equality conjecture. One has

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

Gallardo–Grekos–Pihko conjecture. There is a constant $c$ such that, whenever

Let $\mathcal{A}_3$ and $\mathcal{B}_3$ be the classes of finite abelian groups arising in the paper, and let … while … where $G_1=(\mathbb{Z}/4)^4$,…