24 problems

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…

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…

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

Conjecture. Either $\frak G$ is a Moufang quadrangle, or $\frak G$ is of Clifford type with topological parameters $(3,4k)$ or $(7,8)$.

Cossidente–Siciliano's conjecture. Every special set of $\mathsf{H}(3,q^2)$ is classical, meaning equivalent under the collineation group of $\mathsf{H}(3,q^2)$ to $\mathcal{V}$.

Ghinelli's conjecture. A Singer quadrangle of even order does not exist.

Let $\mathcal{S}$ be a finite generalized quadrangle, and let $p$ be an odd prime. A central symmetry is a point symmetry of the generalized quadrangle, and the group of central sy…

A finite generalized quadrangle is a spherical building of type $\mathsf{B}_2$. A dual hyperbolic line is the dual incidence-geometric analogue of a hyperbolic line, and it is maxi…

Let $\mathbb F$ be an algebraically closed field of characteristic zero, and let $f_2,f_3\in\mathbb F[p_1,l_1]$. Kronenthal–Lazebnik's conjecture. Every graph…

Let $q$ be a prime power, and let a BIBD consist of $q$ copies of a non-Desarguesian affine plane of order $q$. A non-triangular local resolution system is a local resolution syste…

Let $\mathcal O$ be a pointed conic in $\operatorname{PG}(2,q)$, where $q\geq 16$ is even. Non-embedding conjecture. The Tits quadrangle $T_2(\mathcal O)$ is not a proper subquadra…

Swartz's conjecture. The generalized quadrangle ${\mathcal S}$ cannot have a group of automorphisms acting regularly on both its point set and its line set.

Non-extension conjecture. Such an isomorphism

Let $Q$ be a generalized quadrangle of even prime-power order $q$, and let an ovoid be a set of points meeting every line of $Q$ exactly once. Ovoid-intersection conjecture. Every…

Let $\mathcal{Q}$ be a generalized quadrangle of order $s>1$, and suppose a group of automorphisms acts regularly on both the point set and the line set of $\mathcal{Q}$. Nonexiste…

Linearity conjecture. Every translation generalized quadrangle has a projective representation; equivalently, the kernel $k$ is a division ring.

Let $Q(4,q)$ be a parabolic quadric, let an $m$-ovoid be a point set meeting every line of $Q(4,q)$ in exactly $m$ points, and let an elliptic quadric be a three-dimensional ellipt…

Let $Q$ be a generalized quadrangle of order $(q^2,q)$ containing a generalized quadrangle $Q'$ of order $(q,q)$, where $q$ is a power of two. A relative hemisystem is a subset of…

A hemisystem on the Hermitian space $\mathrm{H}(3,q^2)$, for odd $q$, is a set of lines such that every point is incident with exactly half of the lines in the set. Thas's conjectu…

Let $\Gamma^x$ be an infinite translation generalized quadrangle with translation group $T$. Let $z$ be an affine point, meaning a point not collinear with $x$, and let $\{U\}_z$ b…

A skew translation generalized quadrangle (STGQ) is a generalized quadrangle equipped with the relevant skew translation structure; an STGQ is called central if each symmetry about…

Let $(\Gamma,x,E)$ be an STGQ. It has property $(*)$ if property $(*)$ holds at every line incident with its elation point $x$; it is generic if property $(*)$ is not satisfied. In…

Zero-one-all conjecture. Either zero, one, or all lines incident with $x$ have property $(*)$.

Let the three candidate infinite families of hemisystems be the families suggested by patterns in computational data for the classical generalized quadrangles $\mathsf{H}(3,q^2)$.…