18 problems

John–Mitchell conjecture. If an incomplete block design is $D$-optimal, $A$-optimal, or $E$-optimal, then it is an RGD, provided that an RGD exists.

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…

SBISD finiteness conjecture. There are only finitely many $(v,k,2)$-SBISDs.

Let $\mathcal D$ be a $2$-$(v,k,\lambda)$ design with replication number $r$ and $\lambda\geq (r,2\lambda)^2$. An automorphism group of $\mathcal D$ is half-flag-transitive when it…

Paley ETF spark and bender conjecture. The spark of $\Phi_q$ is $((p+1)/2)^s$ and the bender is a $(p^s, ((p+1)/2)^s,\lambda)$-BIBD.

Let $q$ be a prime power, and let a BIBD consist of $q$ copies of a non-Desarguesian affine plane of order $q$. A non-triangular local resolution system is a local resolution syste…

Classification conjecture. Symmetric designs and the Witt designs are the only nontrivial tight designs.

Polynomial-delay code conjecture. There exists a tropical code using the same underlying block design whose maximum delay is polynomial in the relevant parameters, rather than the…

Conjecture on simultaneous A- and D-optimality. Among RGDs, the same design is $A$-optimal and $D$-optimal.

Binder BIBD conjecture. The binder of $\mathcal G(p,\ldots,p)$ is a

Let $K^{(r)}_{n}$ be the complete $r$-graph on $n$ vertices. A block design with given parameters is a design whose underlying hypergraph is $K^{(r)}_{n}$; such a design can exist…

Friendly-family conjecture. If the original design $\mathbb D$ is a Desargues projective plane, then the above construction yields a friendly family of designs.

Let $D=(X,\mathcal{B})$ be a symmetric $2$-$(v,k,\lambda)$ design. For a block $B_0\in\mathcal{B}$, let … be the residual design of $D$, and let $\gamma(D)$ and $\gamma(D_1)$ denot…

Let $D=(X,\mathcal{B})$ be a symmetric $2$-$(v,k,2)$ design with $k\geq 4$, and let $\gamma(D)$ denote its domination number. Domination-number conjecture. Then … This is one of th…

A finite projective plane is a $2$-design whose parameters satisfy the projective-plane axioms; its incidence graph is the bipartite graph joining each point to the blocks containi…

Existence conjecture. For every $v\equiv 1\pmod 6$ with $v\ne 13$, there exists a Steiner triple system of order $v$ with no almost parallel class.

Let $t$ be the number of treatments and $N$ the number of two-block observations. A design is equireplicate when the numbers of times each treatment is tested are as similar as pos…

Threshold conjecture. If $b=cv$ for some constant $c$, then there is a threshold result like the one in Theorem.