License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.50
URN: urn:nbn:de:0030-drops-104541
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10454/
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Mirzaei, Mozhgan ; Suk, Andrew

On Grids in Point-Line Arrangements in the Plane

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LIPIcs-SoCG-2019-50.pdf (0.6 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{mirzaei_et_al:LIPIcs:2019:10454,
  author =	{Mozhgan Mirzaei and Andrew Suk},
  title =	{{On Grids in Point-Line Arrangements in the Plane}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{50:1--50:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10454},
  URN =		{urn:nbn:de:0030-drops-104541},
  doi =		{10.4230/LIPIcs.SoCG.2019.50},
  annote =	{Keywords: Szemer{\'e}di-Trotter Theorem, Grids, Sidon sets}
}

Keywords: Szemerédi-Trotter Theorem, Grids, Sidon sets
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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