4 problems

Solymosi's conjecture. For any set of points $P_0$ and any set of lines $\mathcal{L}_0$ in the plane, the maximum number of incidences between $n$ points and $n$ lines in the plane…

For a finite set $S$ of points in the plane, let $t_k(S)$ be the number of lines meeting $S$ in exactly $k$ points, and let $T_k(S)$ be the number of lines meeting $S$ in at least…

Mirzaei--Suk strengthening. For any set of points $P_0$ and any set of lines $\mathcal{L}_0$ in the plane, there is a constant $\varepsilon=\varepsilon(P_0,\mathcal{L}_0)>0$ such t…

Fix an integer $k\geq 3$. Let $P$ be a set of $n$ points and let $\mathcal{L}$ be a set of $n$ lines in the plane. A $k$-fan is a configuration consisting of distinct points…