23 problems

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…

Cyclic Legendre-pair existence conjecture. Cyclic Legendre pairs exist for every odd $v$.

Let $v$ and $k$ be positive integers, and let a $(\mathbb Z_v,k,1)$-difference family be a collection of $k$-subsets whose differences cover each nonzero element of $\mathbb Z_v$ w…

Let $k$ and $\lambda$ be fixed positive integers with $k\geqslant \lambda+1$. Let $(X,\mathcal B)$ be a cyclic $(v,k,\lambda)$-BIBD with $v\geqslant v_0$, where $v_0$ is an integer…

Let $G$ be an abelian group of order $v$, and let $m>2$. A classical $(v,m,k,\lambda)$-strong external difference family (SEDF) is a family of disjoint $k$-sets…

Let $G=\mathbb{Z}_v$, and let $\mathcal{F}=\{B_1,\dots,B_n\}$ be a cyclic $(v,k,1)$ difference family. A Banff difference family is a disjoint difference family whose blocks avoid…

A 1-rotational Steiner system S$(2,4,v)$ is a Steiner system with block size four admitting an automorphism consisting of one fixed point and a cycle of length $v-1$. On a point se…

Let $G$ be an abelian group, and let an SEDF be a family of pairwise disjoint subsets of $G$ such that, for each set in the family, the external differences between its elements an…

KSWX conjecture. Whenever $x$ is a prime power, there exists a cyclic difference family with the parameter set $$ and multiplier

Existence conjecture. For each D-optimal parameter set $(v;r,s;\lambda)$ there exists a difference family over some abelian group of order $v$ having these parameters.

Symmetry-symbol conjecture. For each normalized and non-exceptional PPS $(v;k_1,k_2,k_3,k_4;\lambda)$ there exist PDFs with symmetry symbols $(s{**})$ and $({**}s)$.

Let $k$ and $\lambda$ be fixed positive integers with $k\geqslant\lambda+1$, and let $(V,\mathcal{B})$ be a cyclic $(v,k,\lambda)$-design. A base block is one chosen block from eac…

Let $(V,\mathcal{B})$ be a cyclic $(v,k,1)$-design: the blocks are partitioned into orbits under a cyclic automorphism, and a base block is one chosen block from each orbit. Disjoi…

Let $C_3^5$ be the elementary abelian group of order $3^5=243$, and let an $(v,m,k,\lambda)$-SEDF denote a strong external difference family with parameters $v$, $m$, $k$, and…

A strong external difference family (SEDF) with parameters $(n,m,k,\lambda)$ in an abelian group is a family of $m$ disjoint $k$-subsets satisfying the external difference conditio…

Complementary exceptional propus-parameter conjecture. In positive integers, this equation has a unique solution $(a,b)$ satisfying $1<a\le(s-1)/2$; moreover, $a-b$ is either a squ…

Prime representation conjecture. Every prime $s\equiv1\pmod{4}$ has a unique representation of this form. The claim arises from the study of the additional exceptional propus param…

Prime-square propus parameter-count conjecture.

Let $v$ be an odd positive integer, and let $\mathbb{Z}_v$ denote the cyclic group of integers modulo $v$. A propus difference family in $\mathbb{Z}_v$ is a cyclic difference famil…

Let $v$ be odd. A propus difference family in $\mathbb{Z}_v$ is a difference family $(X_1,X_2,X_3,X_4)$ with GS-parameters $(v;k_1,k_2,k_3,k_4;\lambda)$ satisfying $k_2=k_3$, with…

Let $G=\mathbb{Z}_v$ be a cyclic group, and let $(X_1,X_2,X_3,X_4)$ be a difference family in $G$ with parameters $(v;k_1,k_2,k_3,k_4;\lambda)$, where … Such a parameter set is cal…

Nonexistence conjecture. There does not exist an $(n,m,k,\lambda)$-SEDF for $m>2$, $k>1$, and $\lambda>1$.

Let $v\equiv 3 \pmod{4}$ be a prime, and let $\mathcal{P}$ be the collection of parameter sets $(v;k_1,k_2,k_3;\lambda)$ satisfying … and … Here an SDS is a supplementary differenc…