On the structure of extremal point-line arrangements
Gabriel Currier, Jozsef Solymosi, Hung-Hsun Hans Yu
TL;DR
This work analyzes near-extremal Szemerédi–Trotter configurations in the plane and proves a strong rigidity phenomenon: if a point set $P$ and line set $L$ determine at least $\Delta|P|^{2/3}|L|^{2/3}$ incidences, then a large subset $P'\subseteq P$ (of size $\Omega(|P|)$) suffices to determine a constant fraction of the arrangement when paired with $L$. The authors combine Guth–Katz polynomial partitioning with the rigidity result of Dvir–Garg–Oliveira–Solymosi, transferring rigidity from collinear triples to the point–line configuration and iteratively recovering most of the lines and points from a fixed core and the incidence data. They introduce a precise rigidity framework, rely on duality between points and lines, and exploit a cell-based decomposition to show that a constant number of rigid cells fixes a large portion of the configuration. The result sharpens the understanding of near-extremal incidence structures and suggests that a substantial part of such configurations can be reconstructed from a small subset of points together with the incidence pattern, with potential implications for characterizing extremal arrangements.
Abstract
In this note, we show that extremal Szemerédi-Trotter configurations are rigid in the following sense: If $P,L$ are sets of points and lines determining at least $C|P|^{2/3}|L|^{2/3}$ incidences, then there exists a collection $P'$ of points of size at most $k = k_0(C)$ such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth-Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester-Gallai configurations.