On Grids in Point-Line Arrangements in the Plane

Abstract

The famous Szemerédi–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 Turán-type result for point-line incidence. Let $\mathcal {L}_a$ and $\mathcal {L}_b$ be two sets of t lines in the plane and let $P=\{\ell _a \cap \ell _b : \ell _a \in \mathcal {L}_a, \,\ell _b \in \mathcal {L}_b\}$ be the set of intersection points between $\mathcal {L}_a$ and $\mathcal {L}_b$. We say that $(P, \mathcal {L}_a \cup \mathcal {L}_b)$ forms a natural $t\times t$ grid if $|P| =t^2$, and ${\text {conv}}P$ does not contain the intersection point of some two lines in $\mathcal {L}_a$ and does not contain the intersection point of some two lines in $\mathcal {L}_b$. 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 $O(n^{{4}/{3}- \varepsilon })$ incidences, where $\varepsilon = \varepsilon (t)>0$. 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 $\Omega ({n^{1+{1}/{14}}})$ incidences.