8 November 2023
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
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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