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DOI: 10.4230/LIPIcs.SoCG.2019.50
On Grids in Point-Line Arrangements in the Plane

Authors: Mozhgan Mirzaei and Andrew Suk

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

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.
Cite as

Mozhgan Mirzaei and Andrew Suk. On Grids in Point-Line Arrangements in the Plane. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 50:1-50:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mirzaei_et_al:LIPIcs.SoCG.2019.50,
  author =	{Mirzaei, Mozhgan and Suk, Andrew},
  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 =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.50},
  URN =		{urn:nbn:de:0030-drops-104541},
  doi =		{10.4230/LIPIcs.SoCG.2019.50},
  annote =	{Keywords: Szemer\'{e}di-Trotter Theorem, Grids, Sidon sets}
}
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