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On dense point-line arrangements in 3D

Published on:

8 November 2023

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Paper Authors:

Andrew Suk,

Ji Zeng


Key Details

Proves k-cliques must exist in dense 3D point-line arrangements with few coplanar lines

Uses polynomial partitioning and the regularity method in the proof

Strengthens our knowledge of extremal behavior for the Guth-Katz theorem

Also gives new bounds on the number of rich points in such arrangements

Provides matching lower bound constructions using a probabilistic argument

AI generated summary

On dense point-line arrangements in 3D

This paper proves that very dense arrangements of points and lines in 3D space, with few lines on a common plane, must contain certain special substructures called k-cliques in general position. This characterizes the extremal behavior for a variant of the Szemerédi-Trotter theorem on incidences between points and lines due to Guth and Katz. The proof uses tools like polynomial partitioning and the regularity method. Overall, this strengthens our understanding of incidence theorems and extremal problems in discrete geometry.

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