130 problems

Determine, up to isomorphism, all simple $5$-$(v,k,\lambda)$ designs with $\lambda=2$ that admit a group of automorphisms acting transitively on their blocks; equivalently, classif…

Determine whether a square integer relative Heffter array $\mathrm{H}_5(n;5)$ exists for every integer $n\ge 8$ satisfying $n\equiv 0\pmod{4}$. The 2026 source claims an affirmativ…

For every integer $n\ge 6$, let $m=\binom{n}{2}$ and consider the incidence family of the complete graph $K_n$, with one column for each vertex and one row for each edge. If…

Given a finite abelian group $G$ of order $v$ and integers $k$ and $\lambda$, determine whether there exists a function $x:G\to\{-1,0,1\}$ such that $\sum_{g\in G}x(g)^2=k$ and, fo…

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…

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 $r\equiv s\equiv t\pmod{2}$,…

Dual Reed–Muller 3-design conjecture. If $q\geq 3$, then for every $\ell\in\mathbb{N}$, $(C^{\perp})_{\ell}$ is not a combinatorial $3$-design.

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…

Daykin–Häggkvist conjecture. Given a partial Latin square $L$ of order $n$ in which each row, column, and symbol is used at most $n/4$ times, it is possible to complete $L$ into a…

Let a $(v,K,\lambda)$-pairwise balanced design be a $v$-vertex hypergraph whose $2$-degree is $\lambda$ and whose edge cardinalities belong to a set of integers $K$. A base block i…

This entry tracks Glock–Kühn–Lo–Osthus, Kelly, and Kwan–Sah–Sawhney–Simkin conjectures. It addresses an unresolved transition regime and sharpens enumeration results for sparse-con…

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

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

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 $T$.

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

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.

Let $\Delta_d$ denote the $d$-point probability simplex, and let a simplex $t$-design be a collection of points in $\Delta_d$ whose moments agree with those of the uniform simplex…

Given $n\ge q>r\ge2$, an $(n,q,r)$-Steiner system is a collection of $q$-element subsets of an $n$-element set in which every $r$-element subset lies in exactly one block. A…

Let $D$ be a design with the parameters of a geometric design $G$ in $\operatorname{PG}(n,q)$ or $\operatorname{AG}(n,q)$, where $q=p^m$. Let the $p$-rank denote the rank of the in…

Folklore conjecture. With high probability,

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…