7 problems

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…

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…

Let $S\subseteq\mathbb{F}_p^3$ be a set of $p^2$ points, where $p$ is prime. The cylinder condition means that $S$ is the union of $p$ pairwise disjoint parallel lines, and hence h…

Let $n\in\mathbb{N}-\{0\}$ and let $K$ be a field. Let $S$ be an affine subspace of $M(n\times n,K)$ such that every element of $S$ is nilpotent, and assume that each element of…

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

Let $[n,m]\rightarrow[k,t]$ mean that every $m$-element subset of $\mathrm{AG}(n,2)$ has a $k$-dimensional affine subspace containing exactly $t$ points, and let $\rho(n;k,t)$ deno…