On cliques in three-dimensional dense point-line arrangements

Abstract

As a variant of the celebrated Szemerédi--Trotter theorem, Guth and Katz proved that $m$ points and $n$ lines in ${\mathbb{R}}^3$ with at most $\sqrt{n}$ lines in a common plane must determine at most $O(m^{1/2}{n}^{3/4})$ incidences for $n^{1/2}\le m\le n^{3/2}$. This upper bound is asymptotically tight and has an important application in Erdős distinct distance problem. We characterize the extremal constructions towards the Guth--Katz bound by proving that such a large dense point-line arrangement must contain a $k$-clique in general position provided $m\ll n$. This is an analog of a result by Solymosi for extremal Szemerédi--Trotter constructions in the plane.