92 problems

Prime-power conjecture for finite projective planes. The order of every finite projective plane is a power of a prime number.

Let $\Pi$ be a finite projective plane admitting a group of automorphisms that acts transitively on its points. Point-transitive projective plane conjecture. The projective plane…

A projective plane is an incidence structure in which every two distinct points lie on a unique line, every two distinct lines meet in a unique point, and there exist four points n…

Fisher's conjecture. The average size of a complete arc is about

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…

A blocking set in a finite projective plane is a set of points meeting every line, and it is small when its size is less than $3(q+1)/2$. A blocking set is linear when it arises fr…

Let $S_0,S_1\subset \mathbb{C}P^2$ be smoothly embedded symplectic surfaces representing the same homology class. Symplectic isotopy conjecture. The surfaces $S_0$ and $S_1$ should…

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…

Kim–Vu conjecture. The constant can be reduced to $C=10$, so that

A projective plane is a finite incidence geometry in which any two points lie on exactly one line, any two lines meet, and there exists a set of four points no three of which lie o…

Saniga's conjecture. If $q$ is not a power of a prime, the question of whether a $q$-dimensional Hilbert space admits a set of $q+1$ mutually unbiased bases is identical to the que…

SPR conjecture. If $d$ differs from a power of a prime number, then the question of whether a $d$-dimensional Hilbert space has a set of $d+1$ MUBs is equivalent to the question of…

Let $PG(2,2^d)$ be the projective plane over ${\mathbb F}_{2^d}$. A monomial hyperoval is a hyperoval projectively equivalent to … The known examples include the regular hyperoval…

A perfect difference set of order $q$ is a set of $q+1$ integers whose pairwise differences, taken modulo $q^2+q+1$, give all nonzero residue classes. The prime power conjecture. A…

Let $q=2^{2k}$, and consider a Hall plane of order $q$ with two conjugate points. Glynn–Stein conjecture. All ovals through the two conjugate points in the Hall plane derive from c…

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

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…

Prime-power conjecture for projective planes. If a projective plane of order $n$ exists, then $n$ must be a prime power.

Uniqueness conjecture. This is the only example of size $s+1$ in any projective plane of order $s$ with that property and such that no flag of the plane is opposite all of its memb…

Let $q=p^n$, with $p$ prime and $n>1$, and let $p^{n/s}$ be the size of the largest proper subfield of ${\mathbb F}_{p^n}$. A $t$-fold blocking set of $\operatorname{PG}(2,p^n)$ is…

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…

Topological breeding conjecture. For every $d\geq 4$, the holomorphic Lefschetz pencil

Conjecture on conic-free untouchable sets. An untouchable set of size $2q+1$ in $PG(2,q)$ may not contain a conic.

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.…