Gabriel Currier, Jozsef Solymosi, Hung-Hsun Hans Yu
{"title":"On the structure of extremal point-line arrangements","authors":"Gabriel Currier, Jozsef Solymosi, Hung-Hsun Hans Yu","doi":"arxiv-2409.06115","DOIUrl":null,"url":null,"abstract":"In this note, we show that extremal Szemer\\'{e}di-Trotter configurations are\nrigid in the following sense: If $P,L$ are sets of points and lines determining\nat least $C|P|^{2/3}|L|^{2/3}$ incidences, then there exists a collection $P'$\nof points of size at most $k = k_0(C)$ such that, heuristically, fixing those\npoints fixes a positive fraction of the arrangement. That is, the incidence\nstructure and a small number of points determine a large part of the\narrangement. The key tools we use are the Guth-Katz polynomial partitioning,\nand also a result of Dvir, Garg, Oliveira and Solymosi that was used to show\nthe rigidity of near-Sylvester-Gallai configurations.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06115","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this note, we show that extremal Szemer\'{e}di-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.