161 problems

Conjecture on the 2-rank of $M_2$.

Let $m\ge 8$ be an even integer, let $b,c\in\mathbb{F}_{2^m}$, and let $r$ satisfy $r\le m/2$ and $\gcd(r,m)=1$. Let $f_{b,c,r}$ denote the function defined in Theorem. Non-APN con…

Let $M$ be a matroid, and call a rank-$k$ flat ordinary if it is the direct sum of a point and a rank-$(k-1)$ flat. Ordinary-flat conjecture. For each integer $k\ge 2$, every compl…

Let $V=\mathbb{F}_q^n$ and let $J_q(n,k)$ be the Grassmann graph whose vertices are the $k$-dimensional subspaces of $V$. A function $f$ on the vertices is Boolean degree $1$ if it…

Let $A_q(n,d;k)$ denote the maximum size of a constant-dimension $k$-subspace code in $\mathbb{F}_q^n$ with minimum subspace distance at least $d$. Generalized vector space partiti…

Bonin's conjecture. The matroid $M$ has at most

Let ${\mathcal D}$ be a symmetric design with Singer parameters … where $q=p^s$ and $p$ is prime. Write ${\rm rank}_p{\mathcal D}$ for the dimension over ${\mathbb F}_p$ of the cod…

Let $\sigma:{\mathbb{F}_q}\rightarrow{\mathbb{F}_q}$ be a permutation, and let $\Gamma_\sigma=\{(x,\sigma(x)):x\in{\mathbb{F}_q}\}\subset{\mathbb{F}_q}^2$ be its graph. A collinear…

For an odd prime power $q$, a Besicovitch set in ${\mathbb{F}_q}^2$ is a subset containing a line in every direction. Finite-plane Besicovitch set conjecture. The smallest Besicovi…

Maximum subgraph conjecture. If $U$ spans a complete subgraph or an empty subgraph of order $q$ in $V_q$, then $U$ is a line in $F_q\times F_q$.

Hollmann's conjecture. If $m$ is an odd prime, then $({\cal E},\{\Delta_a\})$ is pseudocyclic.

Codimension conjecture. The codimension of

Let $n>2$ be prime, and define a permutation $g$ of $\mathbb Z_n$ by … Order permutations lexicographically, and let $\Psi(n)$ be the minimum number of collinear triples in a permu…

Let $q$ be an odd prime power, let $\beta$ be a primitive element of $\mathbb F_{q^2}$, and define … for $r\in\mathbb F_q$. Set … and let $\mathcal U_{\beta}$ be the…

Two-barrier Ore-degree conjecture. If

Prime affine direction conjecture.

The construction associates generators $\rho_1,\rho_2,\rho_3$ to a group $G$ by … A triality construction conjecture. Apart from two cases, up to coloring symmetries, the associate…

Let $V$ be an $n$-dimensional vector space over $\mathbb{F}_q$, and let $m_q(n,k,s)$ be the maximum size of a family $\mathcal{F}\subseteq\genfrac{[}{]}{0pt}{}{V}{k}$ containing no…

Let ${\textup{PG}}(2,q^3)$ be a projective plane with $q$ even and $q>2$. Let $\mathcal{C}$ be a conic with nucleus $N$ tangent to the line $\ell_\infty$ at $T$, and let…

Let $q$ be a prime power. In the strong $(m,n)_q$-Tic-Tac-Toe game on the affine space $4\mathbb{F}_q^m4$, let $T(n,q)$ denote the smallest dimension $m$ for which the first player…

Hegedüs' conjecture. Every such system satisfies

Let $\mathrm{IM}(2,q)$ denote the maximum size of an induced matching in the point-line incidence graph of $\mathbb{F}_q^2$. Prime-field planar induced matching conjecture. There e…

Let $\mathrm{IM}(d,q)$ denote the maximum size of an induced matching in the point-line incidence graph of $\mathbb{F}_q^d$. High-dimensional induced matching conjecture. For every…

Let $A\subseteq\mathbb{F}_p^3$ be a set with $|A|=p$. A determined direction is a direction of a line passing through at least two points of $A$. Directional dichotomy for sets of…

Let $S\subseteq\mathbb{F}_p^3$ be a set of $p^2$ points, where $p$ is prime. The set $S$ is $p$-divisible if every affine hyperplane in $\mathbb{F}_p^3$ intersects $S$ in a number…