282 problems

Hadamard's conjecture. Such a matrix exists whenever $4\mid n$. This is a central existence problem in design theory. The paper notes that order $668$ is the smallest order present…

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…

Let $q$ be a prime power, and let $S_q(t,k,n)$ denote the collection of $q$-Steiner systems with parameters $t,k,n$. A non-trivial $q$-Steiner system has $n\geq 2k$. Metsch's conje…

Euler's conjecture. There are no pairs of mutually orthogonal Latin squares of order $n\equiv 2\pmod 4$.

Let $\mathcal{D}$ be a $3$-$(v,k,\lambda)$ design, and let $G$ be a block-transitive automorphism group of $\mathcal{D}$. The design is point-imprimitive if the action of $G$ on it…

Erdős–Hanani conjecture. Approximate $(n,q,r)$-Steiner systems exist.

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 $n\ge 11$ satisfy $n\equiv 1$ or $3\pmod{8}$, and let $\mathbb{Z}_n$ be the cyclic group of order $n$. A strong Skolem starter is a partition of the nonzero elements of…

A $t$-design is a combinatorial design whose every set of $t$ points lies in the same number of blocks. For a prime power $q=p^s$, a geometric design is the design formed by all…

Erdős' conjecture. For each integer $g\geq 3$ and all sufficiently large $n\equiv 1,3\pmod 6$, there exists a Steiner triple system on $n$ elements with girth at least $g$.

Existence conjecture. Skew-symmetric conference matrices exist for every order divisible by four.

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 $STS(n)$ denote the number of non-isomorphic Steiner triple systems on $n$ vertices, where $n$ is admissible, meaning $n\equiv 1,3\pmod{6}$. Wilson's conjecture. … This gives t…

A strong starter in the cyclic group $\mathbb{Z}_n$ for odd $n$ is a partition of $\{1,2,\dots,n-1\}$ into pairs $\{a_i,b_i\}$ such that the pair sums $a_i+b_i$ are distinct and no…

Let $(X,\mathcal B)$ be a cyclic Steiner triple system of order $v$, with $v\equiv 1\pmod 6$, and identify its point set with $\mathbb Z_v$. Partition the blocks into orbits under…

Lindner's conjecture. Any partial Steiner triple system of order $m$ can be embedded in a Steiner triple system of order $n$ whenever

Let a simple $\operatorname{T}(t,k,v)$ trade be a trade on a $v$-element set in which no block is repeated. A $(t,k,v)$-halving is a simple $\operatorname{T}(t,k,v)$ trade with vol…

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.

Wanless's conjecture. Every partite $(d,d+1,n)$-Steiner system has a perfect matching if $n$ or $d$ is even.

Let $K_n^r$ be the complete $r$-uniform hypergraph and $K_q^r$ its complete $q$-vertex subhypergraph. A packing is a collection of edge-disjoint copies of $K_q^r$ in $K_n^r$. Erdős…

Let $I$ and $J$ be positive integers. A regular $(I+1,J,(I+1)J)$-DGR is a collection of $I+1$ disjoint $J$-mark Golomb rulers partitioning $\{1,\ldots,(I+1)J\}$. Regular extension…

Ionin–Shrikhande conjecture. For any Ryser design on $v$ points,

Folklore conjecture. With high probability,

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…