30 problems

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…

A blocking set of a finite linear space $\mathsf{S}$ is a subset $B$ of $\mathsf{S}$ such that each line of $\mathsf{S}$ contains a point of $B$ and a point of…

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…

Kriesell's conjecture. The projective plane $\mathbb{P}^2(\mathbb{F}_q)$ can be partitioned into

Cubic-curve extremal conjecture. If $|B|=n$, then all points in $P\cup B$ lie on a cubic curve.

Erdős–Purdy conjecture. The number of blocking points required is at least $(1+o(1))n$.

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 ${\mathcal B}$ be a double blocking set of size $3q-1$ in $\operatorname{PG}(2,q)$, and let an $(q-1)$-secant be a line meeting ${\mathcal B}$ in $q-1$ points. Hill's conjectur…

Let $q$ be a prime power, and let a triangle in $\operatorname{PG}(2,q)$ be given. Remove $x$ points from its sides and add $x-1$ points, obtaining a set of $3q-1$ points. Hill's c…

Szőnyi's conjecture. The blocking set $B$ has at least

Let a minimal $t$-fold blocking set be a minimal point set in a projective plane such that every line meets it in at least $t$ points. Suppose the projective plane has prime power…

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 $q^t=p^h$, where $p$ is prime. A small minimal blocking set in the projective space $\mathrm{PG}(n-1,q^t)$ is a minimal blocking set satisfying the usual smallness condition, a…

Doyen's conjecture. The least possible size of a blocking set in $\mathrm{AG}(n,q)$ is

Small minimal blocking set linearity conjecture. Every small minimal blocking set should be an $\mathbb{F}_q$-linear set.

Let ${\mathscr C}$ be an ideal minimally non-packing clutter, meaning an ideal clutter that does not have the packing property but whose proper minors have the packing property. It…

Let $q=p^t$ be a prime power, and let $B$ be a small minimal blocking set in $\mathrm{PG}(n,q)$, meaning that $B$ meets every hyperplane, satisfies $|B|<3(q+1)/2$, and has no prope…

Let $s$ and $t$ be integers satisfying … For the second weight formula, let $c_t$ be defined by the condition that $c_t+(q-t)q$ is the second weight of the code…

Linearity conjecture. All small minimal $k$-blocking sets in $\mathrm{PG}(n,q)$ must be linear.

Let $P$ be a $k$-blocked point set, with its points assigned one of $k$ colours so that two distinct points have the same colour exactly when some other point of $P$ blocks them. C…

A finite point set in the plane is $k$-blocked if its points are assigned one of $k$ colours such that two distinct points have the same colour exactly when some other point of the…

Let $b(n)$ be the minimum integer such that some set of $n$ points in the plane in general position is blocked by a set of $b(n)$ points. Superlinear blocker-number conjecture. … K…

A finite point set $P$ in the plane is $k$-blocked if each point is assigned one of $k$ colours, and two distinct points are assigned the same colour exactly when some other point…

For fixed $\ell$, let $b_\ell(n)$ be the minimum integer such that every set of $n$ points in the plane with no $\ell$ collinear points can be blocked by a set of $b_\ell(n)$ point…

A finite point set is in convex position if every point is a vertex of its convex hull. A blocker for such a set is a point, outside the set, lying on an edge segment determined by…