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arXiv · 2311.04804

On cliques in three-dimensional dense point-line arrangements

Andrew Suk·Ji Zeng
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Abstract

As a variant of the celebrated Szemer\'edi--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 $n^{1/2}\leq m\leq n^{3/2}$. This upper bound is asymptotically tight and has an important application in Erd\H{o}s 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 Szemer\'edi--Trotter constructions in the plane.

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Andrew Suk, Ji Zeng. 2023-11-08. On cliques in three-dimensional dense point-line arrangements. https://arxiv.org/abs/2311.04804

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