28 problems

Prefix sum recurrence conjecture. For an absolute constant $A>0$, independently of $a\in\mathbb{F}_p$,

A smoothly embedded projective plane in $S^4$ is knotted if it is not the standard unknotted embedding, and it is reducible if it admits an unknotted summand that is not a $2$-sphe…

Let $\Pi$ be a projective plane of order $q$, let $U$ be a rank-$3$ residue zero-pattern model satisfying the identity-pattern minor constraints, and let $T_U$ denote its tangent s…

Let $q\geq 3$, let $\Pi$ be a projective plane of order $q$, and consider a $4\times4$ determinant initial form of its combinatorial incidence matrix. A degenerate diamond is a con…

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 projective plane has an integer $n$, called its order, such that each line contains $n+1$ points, each point lies on $n+1$ lines, and there are $n^2+n+1$ points and lines.…

Let $q$ be a prime power and let ${\mathcal U}$ be a non-classical unital in ${\textup{PG}}(2,q^2)$. A Triple O'Nan configuration is a set of six distinct lines pairwise intersecti…

Fano-plane conjecture. Every non-classical unital ${\mathcal U}$ in ${\operatorname{PG}}(2,q^2)$ contains a Fano plane if $q\geq 4$.

Extremal homology-manifold conjecture. 1. Suppose that $K$ is a $15$-vertex $Z$-homology $8$-manifold such that

Projective-plane extremal conjecture. For $m\geqslant r+2$,

Let $D\subset \mathbb{Z}/m\mathbb{Z}$ be a perfect difference set if every nonzero $a\in\mathbb{Z}/m\mathbb{Z}$ can be written uniquely as the difference of two elements of $D$. A…

Let $D$ be a design associated with a maximal arc of degree $4$ in $PG(2,2^m)$, where $m\ge 2$, and let $A$ be its incidence matrix. Let $C$ be the binary code spanned by the rows…

Let $\Pi$ be a projective plane of order $p^s$, where $p$ is prime, and let its incidence matrix have rank over $\mathbb{F}_p$ called its $p$-rank. Hamada–Sachar conjecture. The…

A minimally nonideal matrix is a clutter matrix that is not ideal but whose proper minors are all ideal. Its core is the submatrix consisting of its minimum-weight rows. A non-dege…

Let $q$ be a prime power satisfying $q\geq 13$ and $q\not\equiv 2\pmod 3$. A double blocking set in $\operatorname{PG}(2,q)$ is a set meeting every line in at least two points; a…

Let $\pi$ be a projective plane of prime order $p$, and let $C_p^\perp(\pi)$ denote the dual of its $p$-ary code. For a codeword, its Hamming weight is the number of nonzero coordi…

A projective plane $\sigma$ is a subplane of a projective plane $\pi$ if $\sigma$ is a subsystem of $\pi$. A projective plane is prime if it has no proper projective subplane. Let…

Prime-power conjecture. If a finite projective plane of order $n$ exists, then $n$ is a power of some prime $p$.

Projective-plane coclique conjecture. For $q>2$, a coclique of maximum size in the graph $\Gamma_q(q^2+q+1,q+1)$ must consist of all the $(q+1)$-sets containing two given points; s…

The $\sqrt{q}$ conjecture. The second largest complete arc in ${\mathrm{PG}}(2,q)$ has size

Let $t\geq 2$, and let $\mathcal{D}$ be a $2-(2^{2t-1}-2^{t-1},2^{t-1},1)$ design with incidence matrix $A$. Carpenter–Tonchev's rank conjecture. The binary rank of $A$ satisfies ……

Neumann's arc-count conjecture. The following equivalent inequalities should hold: for every finite non-Desarguesian plane,

Baer-subplane covering conjecture. If $\mathcal{L}$ covers every point of $\Pi_q$ outside $\Pi'$, then $\mathcal{L}$ is a covering set; in particular, it also covers every point of…

Let $PG(2,q)$ be the projective plane of order $q$, and let $t_{2}(2,q)$ denote the smallest size of a complete arc in $PG(2,q)$. The finite-range upper-bound conjecture. In…

Let $PG(2,q)$ be the projective plane of order $q$, and let $t_{2}(2,q)$ denote the smallest size of a complete arc in $PG(2,q)$. The logarithmic upper-bound conjecture. In…