48 problems

Let $j$ and $k$ be nonnegative integers with $k$ even. Prescribed-defect existence conjecture. There exists a positive integer $n$ and a set $S\subset\{0,1,\dots,n-1\}$ such that…

For $n\ge 1$, let $S$ range over subsets of $\{0,1,\dots,n-1\}$. For nonnegative integers $j$ and $k$, define … assuming the limit exists. Since $S-S$ is symmetric about $0$, only…

Limiting-proportion conjecture. The three limiting proportions all exist and are positive.

Let $p$ be a prime, and let $\mathcal{R}_p$ denote the set of nonzero squares in ${\mathbb F}_p$. Lev and Sonn's conjecture. For all sufficiently large primes $p$, there is no subs…

Let $p$ be prime, and let $K$ be the multiplicative subgroup of $\ell$-th powers in $\mathbb{F}_p$. A power difference set conjecture asserts that if $K$ forms a difference set of…

A cyclic projective plane is a projective plane admitting a regular action by a cyclic group. A projective plane is Desarguesian when it satisfies Desargues's theorem, and a perfec…

A finite Sidon set is a finite set of integers whose differences of distinct elements are all distinct. A finite perfect difference set is a finite set whose differences represent…

Let $p$ be an odd prime and let $g$ be a natural number. Define … Theorem 1 establishes that these limits exist and are positive real numbers. Even–odd limit conjecture. The two li…

Let $G$ be a finite group, and let an ADS be obtained from a difference set in $G$ by adding or removing one element. The one-element ADS conjecture. If $G$ is cyclic or has odd or…

A cyclic 2-design is a finite unitary orbit whose associated decohered probability vectors form a simplex 2-design. Infinite-family conjecture. Cyclic 2-designs exist in an infinit…

Let $s$ be a positive integer with $s-1$ a prime power, let $S$ be a perfect difference set of size $s$, and let $G_m(S)$ be the graph constructed in Corollary. A bipartite graph i…

Let $a=(a_n)_{n\in\mathbb N}\in(0,1)^{\mathbb N}$, and let $C(a)$ denote the central Cantor set determined by the sequence $a$. Write $\operatorname{int}$ for interior in…

Let $b$ be a positive integer, and let $g(n,k,\ell)$ denote the relevant extremal function for difference sets. For the recursively constructed $b$-ary projected-hypercube sets, th…

Let $d \geq 2$ and let $A \subset \mathbb{R}^d$ be a finite set with affine dimension $\dim(A)=d$. Stanchescu's conjecture. Then … This conjecture proposes that Stanchescu's constr…

Tait's perfect-difference-set extension conjecture. The set $S$ is contained in some perfect difference set $S'$ of size $p+1$.

Type 2 existence conjecture. Legendre difference families of type $2$ exist for all odd lengths $v>8$.

Power-of-2-or-3 conjecture. If $\mathcal{S}$ is a $pg(s,t,\alpha)$ of rigid type, then

Non-existence conjecture. Let $m\geq 5$ be a prime, $\gamma_1,\gamma_2\in\mathbb Z$, and $n\in\mathbb Z^+$. There does not exist a non-trivial almost $m$-ary NPS of type…

No-divot conjecture. The limiting probabilities satisfy

Peak-at-four conjecture. For all $n\ge 15$ and every $k\ne4$,

Divot persistence conjecture. If $k$ is a divot of $P_{n_1}^{\mathrm{H}}$, then $k$ is also a divot of $P_n^{\mathrm{H}}$ for every $n>n_1$.

Divot classification conjecture. For every $k>1$, $k(k-1)+3$ is a divot of $P_n^{\mathrm{H}}$ for sufficiently large $n$. Furthermore, these are the only divots.

Let $D$ be a $(v,k,\mu)$ difference set in a group $G$, and define … for $1\leq i\leq\gcd(k-1,\mu)$. A graph is $D$-graceful when its vertices can be labeled by the elements of $D$…

Let $m\geq 3$ be odd, and define $k_m$ recursively by … For $b,c\in {\mathrm{GF}}(3^m)$, let $B_{(x^{10},b,c)}$ denote the corresponding block, and let ${\mathcal{B}}_{(x^{10},k)}$…

Reversible Hadamard difference set enumeration conjecture. The total number of reversible Hadamard difference sets in $G_n$ of type A is