98 problems

Let $G=(G,+)$ be a finitely generated discrete abelian group. For subsets $A,F\subseteq G$, write $A\oplus F=G$ when every element of $G$ has a unique representation $a+f$ with…

Let $G$ be a finite abelian group, and let $\mathcal{F}(G)$ be the free abelian monoid on $G$. Define … and let … be the block monoid on $G$. The Betti elements of an affine monoid…

Let $(X,r)$ be an involutive solution of size $n$, and let $g(n)$ denote the bound used in the surrounding text for solutions whose permutation group is abelian. Abelian permutatio…

Ruzsa's conjecture. If there exists a constant $K$ such that

Let $V$ be a finite tile in $\mathbb{Z}^n$. A tiling by translations is fully periodic if its translation set has a finite-index period subgroup. Lagarias--Wang conjecture. If $V$…

A reflexive abelian group is an abelian group naturally isomorphic to its double dual. Existence conjecture. There are reflexive abelian groups of arbitrarily large cardinalities.…

Strengthened 2-criticality conjecture. The set $\operatorname{gcs}(\alpha_{k,k'}L_s)$ is 2-critical and strong, and completes top down to $\alpha_{k,k'}L_s$.

Simultaneous double product property conjecture. For arbitrarily large $n$, there exists an abelian group $H$ with $n$ pairs of subsets $A_i,B_i$ satisfying the simultaneous double…

Pyber's conjecture. There exists a fixed constant $c>1$ such that

Let $G$ be the abelian group under discussion, and call a subset saturated if it is a maximal avoidable set. The sets described above are those obtained as solution sets of … where…

Let $(\Gamma,g,G)$ be an abelian gain graph, where $G$ is a torsion-free abelian group, possibly $2$-divisible. Let $Z_1(\Gamma)$ be the integral cycle group, and suppose it has a…

Let $G$ be a finite abelian group with $|G|>1$, and let $p$ be the smallest prime divisor of $|G|$. Let $k<p$, let $a_1,\ldots,a_k\in G$ be pairwise distinct, and let…

Cyclic interval conjecture. The displayed inequality holds for every odd $m$ and every nonempty proper subset $x\subseteq\mathbb{Z}_m$; consequently,

Let $G$ be an abelian group, and let $A$ be a finite subset of $G\setminus\{0\}$. An ordering $a_1,\ldots,a_{|A|}$ of $A$ is sequenceable if its partial sums … are pairwise distinc…

Let $t$ be a positive integer, let $G$ be an abelian group, and let $A,B\subseteq G$ be finite subsets with $|A|,|B|\geq t$. Write $A+_iB$ for the set of elements of $G$ having at…

Let $G$ and $H$ be finite abelian groups of the same order. Define the sum-of-element-orders function by … where $o(g)$ denotes the order of $g$. Tărnăuceanu's conjecture. The grou…

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

Let $\Gamma$ be a finite abelian group with generating set $S$, and let $G=\operatorname{Cay}(\Gamma,S)$. Let $\mathcal{H}$ denote the class of graphs having a unique Hamiltonian c…

Let $G$ be a type $I$ abelian group of order $n$. Write $f^\star_{\max}(G)$ for the number of maximal distinct sum-free subsets of $G$, and let $\mu(G)$ denote the maximum density…

Let $k\bin\mathbb N$ and set $n:=5^k$. Write $f_{\max}(G)$ for the number of maximal sum-free subsets of a finite abelian group $G$. The elementary abelian 5-group conjecture. … Th…

Let $k\in\mathbb{N}$. Let $\Gamma$ be a countably infinite abelian group, and suppose … For $A\subseteq\Gamma$, write $d^*(A)$ for its upper Banach density, and for a set…

Abelian embedding conjecture. For a distribution $\mu$ on $\Sigma^k$, Conclusion $$ holds if and only if $\mu$ admits no Abelian embedding.

Let $H$ be a subgroup of $\mathbb{Q}$, and consider a computable group of the form … A computably simple-copy conjecture. Theorem holds for any computable group of this form. That…

Following Aliabadi, an abelian group $G$ has the weak acyclic matching property if, for every pair of finite nonempty subsets $A,B\subseteq G$ with $|A|=|B|$ and…

The condensed Whitehead conjecture. Suppose that $A$ is an abelian group, $\mathbb{B}$ is a complete Boolean algebra, and