On cliques in three-dimensional dense point-line arrangements

2023/11/08 by Suk, Andrew, Zeng, Ji
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.04804

Abstract

As a variant of the celebrated Szemerédi--Trotter theorem, Guth and Katz proved that m points and n lines in ℝ³ with at most √(n) lines in a common plane must determine at most O(m^1/2n^3/4) incidences for n^1/2≤ m≤ 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 ≪ n. This is an analog of a result by Solymosi for extremal Szemerédi--Trotter constructions in the plane.

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