文章基本信息

标题：On Grids in Point-Line Arrangements in the Plane
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作者：Mozhgan Mirzaei ; Andrew Suk
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2019
卷号：129
页码：1-11
DOI：10.4230/LIPIcs.SoCG.2019.50
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：The famous Szemerédi-Trotter theorem states that any arrangement of n points and n lines in the plane determines O(n^{4/3}) incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let L_a and L_b be two sets of t lines in the plane and let P={l_a cap l_b : l_a in L_a, l_b in L_b} be the set of intersection points between L_a and L_b. We say that (P, L_a cup L_b) forms a natural t x t grid if P =t^2, and conv(P) does not contain the intersection point of some two lines in L_a and does not contain the intersection point of some two lines in L_b. For fixed t > 1, we show that any arrangement of n points and n lines in the plane that does not contain a natural t x t grid determines O(n^{4/3- epsilon}) incidences, where epsilon = epsilon(t)>0. We also provide a construction of n points and n lines in the plane that does not contain a natural 2 x 2 grid and determines at least Omega(n^{1+1/14}) incidences.
关键词：Szemerédi-Trotter Theorem; Grids; Sidon sets