125 problems

For positive integers $r\le s\le t$, the complete tripartite graph $K_{r,s,t}$ admits a decomposition into $5$-cycles if and only if $r\equiv s\equiv t\pmod{2}$,…

Fix an integer $t\geq 3$. For a $t$-element set $A\subset\mathbb{Z}$, let $\Delta(A)=\{a_j-a_i:a_i,a_j\in A,\ a_i<a_j\}$ be its set of positive differences. Call $A$ a Golomb ruler…

Let $\widetilde{\mathscr{D}}$ be a design and let ${\mathscr{D}}(\widetilde{\mathscr{D}},(e_1,e_2))$ denote the design obtained from it by the rectangular combination construction,…

Let $K_v^{(3)}$ be the complete $3$-uniform hypergraph on vertex set $\mathbb{Z}_v$, and let $I$ consist of the missing triplets $(i,i+v/3,i+2v/3)$ whenever $3\mid v$. A decomposit…

Classification conjecture. Then $d=2,3,6,7,23$, and $\Omega$ is isometric to the corresponding example found in the examples section.

Polynomial-size uniform subset conjecture. There exists an $X$-uniform subset $T\subset G$ such that

Let ${\mathcal D}$ be a symmetric design with Singer parameters … where $q=p^s$ and $p$ is prime. Write ${\rm rank}_p{\mathcal D}$ for the dimension over ${\mathbb F}_p$ of the cod…

Let $\Delta$ be an $l$-CM simplicial complex of maximal girth. Let $c$ denote the parameter used for its dimension in the paper, and suppose that $\Delta$ is not the $r$-skeleton o…

Exact transitive-triple packing conjecture.

Let $\widetilde{\mathscr{D}}$ be the input design, let $e$ and $\widetilde{k}$ be the parameters of Construction $\mathscr{D}(\widetilde{\mathscr{D}},e)$, and let…

Erdős meets Nash-Williams' conjecture. For every integer $g$, every sufficiently large $K_3$-divisible graph $G$ on $n$ vertices satisfying

Nash-Williams' conjecture. If $G$ is $K_3$-divisible and

Let $v,k,\ell\in\mathbb{Z}^+$ satisfy the necessary parameter equation referred to in the source as Equation, and suppose that there exists a symmetric BIBD on $v$ points with bloc…

A Hadamard matrix is a square array with entries in $\{+1,-1\}$ such that every two distinct rows agree in exactly half of their columns. Hadamard's conjecture. A Hadamard matrix o…

Let $C$ be a PULB-optimal code, and let $D$ be its set of universal minima. Assume that $D$ is in general position, meaning that $D$ is not contained in a hyperplane. Universal pol…

Agrawal's ordering conjecture. For any $(r\times c,r+c-1)$-unordered triple array $U$ with $e>2$, there is an $(r\times c,r+c-1)$-triple array $T$ with $U_T=U$.

Agrawal's conjecture. If there is a symmetric $2$-$(r+c,r,d706_{cc})$ design with $r-d706_{cc}>2$, then there is an $(r\times c,r+c-1)$-triple array.

Embedding conjecture. If $H$ is a regular Hadamard matrix of order $v^2$, then such a containing cube and subcube exist.

Glock–Kühn–Osthus conjecture. For each integer $r\ge 3$, there exists a constant $C>0$ such that, for all integers $q>r$ and all sufficiently large $n$, every $K_q^r$-divisible $r$…

Let $[n]=\{1,2,\ldots,n\}$, let $\mathscr{P}(n)$ be the power set of $[n]$, and let a family be an $(a,b)$-town modulo $k$ when all its sets have cardinality congruent to $a$ modul…

Cavenagh–Hämäläinen–Lefevre–Stones' conjecture. For each $r,\lambda\in\mathbb N$, there exists an integer $n(r,\lambda)$ such that, for every $n\geq n(r,\lambda)$, every simple par…

Gronau–Mullin–Rosa conjecture. For every $n$-vertex tree $T$ other than the path on $4$ vertices, $K_n$ has an orthogonal double cover by copies of $T$.

Let $G$ be a quaternionic reflection group, or more generally a finite irreducible group of $d\times d$ matrices over the quaternions $\mathbb{H}$. A subgroup is reducible if it pr…

Partite folklore conjecture. For each integer $q\geq 4$, the following holds for sufficiently large $n$: if $G$ is a balanced partite-$K_q$-divisible $q$-partite graph on $qn$ vert…

Partite Nash-Williams conjecture. For sufficiently large $n$, if $G$ is a balanced partite-$K_3$-divisible $3$-partite graph on $3n$ vertices with