215 problems

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:

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…

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

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…

Let $q$ be a number, and let $G$ be an Abelian group with … A subset $B\subset G$ is a difference set if every non-unit element of $G$ has a unique representation $g=ab^{-1}$ with…

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 $\Gamma$ 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…

Let $a,b>1$, let $(\Gamma,+)$ be an abelian group, and let $\Upsilon$ denote the set of all finite abelian groups of odd order or containing more than one involution. An…

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 $G$…

Sandpile group distribution conjecture. For every finite abelian group $G$,

Let $G$ be a finite abelian group of order $n$, let $p$ be the smallest prime divisor of $n$, and let $\rho\hat{\;} (G,m,h)$ denote the minimum restricted $h$-fold sumset size amon…

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 with $|G|=ph$, where $p\geq3$ is the smallest prime divisor of $|G|$ and $h$ is composite. Let $c(G)$ denote the critical number associated with i…

Let $A$ be an incomplete subset of $\mathbb{Z}_p$ with $|A|\geq\sqrt{2p}$. After a proper dilation, suppose the Deshouillers–Freiman structural estimate is considered. Deshouillers…

Let $G$ be a finite Abelian group and $T\subset G$ a tile. Suppose that $S\subset\widehat G$ satisfies … where $Z_T$ is the Fourier zero-set of $T$. Such an $S$ is a universal spec…

Let $G$ be a finite Abelian group, and let $T\subset G$ be a tile. A subset $S\subset\widehat G$ is a universal spectrum of $T$ if $S$ is a spectrum of every tiling complement $T'$…

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 $G$ 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 $n>1$ 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…

Hamidoune–Plagne conjecture. The largest possible density is