# On grids in point-line arrangements in the plane

## TL;DR

This paper extends the Szemerédi-Trotter theorem by showing that arrangements avoiding a specific grid structure have fewer incidences, and constructs examples with many incidences that avoid such grids.

## Contribution

It introduces a Turán-type bound for point-line arrangements avoiding a natural grid, and provides explicit constructions with high incidences without such grids.

## Key findings

- Arrangements without a natural t×t grid have O(n^{4/3 - ε}) incidences.
- Constructed arrangements without a 2×2 grid have at least Ω(n^{1+1/14}) incidences.
- The results generalize incidence bounds by excluding specific grid configurations.

## Abstract

The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for point-line incidence. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be two sets of $t$ lines in the plane and let $P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\}$ be the set of intersection points between $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $(P, \mathcal{L}_1 \cup \mathcal{L}_2)$ forms a \emph{natural $t\times t$ grid} if $|P| =t^2$, and $conv(P)$ does not contain the intersection point of some two lines in $\mathcal{L}_i,$ for $i = 1,2.$ For fixed $t > 1$, we show that any arrangement of $n$ points and $n$ lines in the plane that does not contain a natural $t\times t$ grid determines $O(n^{\frac{4}{3}- \varepsilon})$ incidences, where $\varepsilon = \varepsilon(t)$. We also provide a construction of $n$ points and $n$ lines in the plane that does not contain a natural $2 \times 2$ grid and determines at least $\Omega({n^{1+\frac{1}{14}}})$ incidences.