418 problems

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

Kantor's conjecture. Every finite flag-transitive generalized quadrangle is classical, isomorphic to $\mathrm{GQ}(3,5)$, or is the generalized quadrangle of order $(15,17)$ arising…

Let $\mathrm{PG}(3,q)$ be three-dimensional projective space over $\mathbb{F}_q$. A Cameron–Liebler line class is a set of lines meeting every spread in a fixed number $x$ of lines…

Let $q$ be a prime power, and let $S_q(t,k,n)$ denote the collection of $q$-Steiner systems with parameters $t,k,n$. A non-trivial $q$-Steiner system has $n\geq 2k$. Metsch's conje…

An O'Nan configuration in a unital is a collection of four distinct lines meeting in six distinct unital points. Piper's conjecture. Any unital without an O'Nan configuration is th…

Let $q=p^n$ be a prime power with characteristic $p>3$, and let $X^d$ be a monomial over $\mathbb{F}_q$. A monomial is planar if its associated function has the planar property. De…

A $t$-design is a combinatorial design whose every set of $t$ points lies in the same number of blocks. For a prime power $q=p^s$, a geometric design is the design formed by all…

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…

Let $q$ be an odd prime power, let $\mathbb{F}_{q^2}$ be the finite field with $q^2$ elements, and let $A\subseteq\mathbb{F}_{q^2}$ satisfy $0,1\in A$ and $|A|=q$. Suppose that…

Niederreiter's conjecture. The number of $m$-dimensional $\alpha$-splitting subspaces is

Clifton–Huang conjecture. For $k \ge 2$ and $n$ sufficiently large,

Let $q\geq 5$ be a prime power, with the corresponding parity and residue value $\xi\in\{-1,0,1\}$ determined by $q\equiv\xi\pmod 3$. Let $G_q$ act on the EnΓ-lines in…

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

Edoukou–Ling–Xing conjecture. The maximum is

Hegedüs' conjecture. Every such system satisfies

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 $q$ be a prime power, and let the collinearity graph of the polar space $\mathcal{Q}^{-}(5,q)$ be the graph whose vertices are the points of $\mathcal{Q}^{-}(5,q)$, with two ve…

Let $q$ be a power of $3$, let $G=PGL_2(q)$ act on the lines of $PG(3,q)$, and let $C$ be the twisted cubic. A line is generic if it neither intersects $C$ nor lies in any osculati…

Kovács–Nagy density conjecture. For every $k\ge 1$ and $0\le t\le 2^k$, we have

Let $q=p^e$ be an odd prime power and let $k$ be an integer with $1\le k\le q-1$. Define … and … as polynomials in $\mathbb F_q[X]$. Dmytrenko–Lazebnik–Williford's conjecture. Each…

For $q=2^e$, consider the generalized quadrangles arising from monomial graphs over $\mathbb F_q$. Glynn's conjecture. The known examples of such generalized quadrangles represent…

For $d\in\mathbb{N}$, consider arrangements of distinct points in $\mathbb{CP}^{d-1}$, with distance measured by the Fubini--Study metric. Zauner's conjecture. For every…

Nonexistence conjecture. The polar space $\mathsf{Q}^+(9,q)$ does not have an ovoid.

Ghinelli's conjecture. The generalized quadrangle ${\mathcal S}$ does not admit a group of automorphisms acting regularly on its point set.