184 problems

Let $G$ be an abelian group, and let $A=\{a_1,\ldots,a_k\}$ and $B=\{b_1,\ldots,b_k\}$ be subsets of $G$, where $G$ has odd order. Snevily's conjecture. There is a permutation…

A Hamilton decomposition of a graph is a decomposition of its edge-set into Hamilton cycles. Alspach's conjecture. Every abelian Cayley graph of even degree admits a decomposition…

Let Euclidean space of any dimension be tiled by hypercubes whose positions lie in a lattice. Minkowski's conjecture. Some pair of hypercubes must meet face-to-face. This is the ge…

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 $d>1$ and let $cOmegacsubset cmathbb{Z}^d$ be a finite set. Lagarias–Wang's conjecture. If $cOmega$ tiles $cmathbb{Z}^d$ by translation, then it admits a periodic tiling. Here…

Let $G$ be an abelian group and let $S\subseteq G$ be a subset not containing the identity element $0$. The set $S$ is sequenceable if its elements can be ordered as…

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 $C_n^r$ 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…

Erdős–Lemke conjecture. Every such sequence contains a zero-sum subsequence whose sum is at most $\operatorname{lcm}(n,a_1,\ldots,a_n)$.

Let $(G,\cdot)$ be a group. A subset $S$ of the non-identity elements of $G$ is sequenceable if its elements can be ordered so that the associated partial sums are either all disti…

Let $G$ be an abelian group, let $m$ be a positive integer, and let $L$ denote the subgroup of involutions in $G$. Write $\rho\hat{\;} (G,m,2)$ for the minimum size of a restricted…

Let $\Gamma$ be an abelian group and let $\mathcal{C}\subseteq\Gamma$ be a connection set not containing the identity. Let $Cay(\Gamma,\mathcal{C})$ be the Cayley graph on $\Gamma$…

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

Costa–Morini–Pasotti–Pellegrini conjecture. For any abelian group $(G,+)$ and any subset $A\subseteq G\setminus\{0\}$ such that there is no $x\in A$ with $\{x,-x\}\subseteq A$ and…

Let $(W,B)$ be a spherical latin bitrade, with $\mathcal{C}_W$ and $\mathcal{C}_B$ denoting the finite torsion groups associated with the partial latin squares $W$ and $B$, respect…

Banks–Pappalardi–Shparlinski conjecture. The “very split” groups, for which $n_1$ is very large compared with $n_2$, occur with density zero; equivalently,

Let ${\mathbb Z}_{(\rho)} \subseteq {\mathbb Q}$ be the localization appearing in the preceding completeness result, and consider non-free $n$-free-by-$1$ groups in its place. Comp…

Three-generator optimality conjecture. The graphs given by that theorem are the largest undirected Cayley graphs of Abelian groups on three generators for each diameter $k$.

Three-generator Abelian Cayley graph conjecture. The constructed graph is as large as possible among undirected Cayley graphs of Abelian groups on three generators with diameter…

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 $\alpha_{k,k'}L_s$.

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…

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…

Mycielski's conjecture. For every $t=1,\ldots,k$, one has