Planar point sets determine many pairwise crossing segments

Abstract We show that any set of n points in general position in the plane determines n 1 − o ( 1 ) pairwise crossing segments. The best previously known lower bound, Ω ( n ) , was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

[1]  János Pach,et al.  A Ramsey-type result for convex sets , 1994 .

[2]  János Pach,et al.  Research problems in discrete geometry , 2005 .

[3]  Noga Alon,et al.  Crossing patterns of semi-algebraic sets , 2005, J. Comb. Theory, Ser. A.

[4]  István Tomon.  Turán-Type Results for Complete h-Partite Graphs in Comparability and Incomparability Graphs , 2016, Order.

[5]  L. Mirsky.  A Dual of Dilworth's Decomposition Theorem , 1971 .

[6]  János Pach,et al.  Density and regularity theorems for semi-algebraic hypergraphs , 2015, SODA.

[7]  Pavel Valtr.  Lines, line-point incidences and crossing families in dense sets , 1996, Comb..

[8]  Haim Kaplan,et al.  Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique , 2011, Discret. Comput. Geom..

[9]  Eyal Ackerman.  On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges , 2009, Discret. Comput. Geom..

[10]  Csaba D. Tóth,et al.  Intersection patterns of curves , 2011, J. Lond. Math. Soc..

[11]  János Pach,et al.  String graphs and incomparability graphs , 2012 .

[12]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[13]  L. Guth,et al.  On the Erdős distinct distances problem in the plane , 2015 .

[14]  J. Pach,et al.  A semi-algebraic version of Zarankiewicz's problem , 2014, 1407.5705.

[15]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

[16]  János Pach,et al.  Overlap properties of geometric expanders , 2011, SODA '11.

[17]  Jean Cardinal,et al.  The Clique Problem in Ray Intersection Graphs , 2013, Discret. Comput. Geom..

[18]  Gábor Tardos,et al.  On the maximum number of edges in quasi-planar graphs , 2007, J. Comb. Theory, Ser. A.

[19]  F. Thomas Leighton,et al.  Complexity Issues in VLSI , 1983 .

[20]  Micha Sharir,et al.  Quasi-planar graphs have a linear number of edges , 1995, GD.

[21]  Larry Guth,et al.  Algebraic methods in discrete analogs of the Kakeya problem , 2008, 0812.1043.

[22]  Pavel Valtr.  On Mutually Avoiding Sets , 2013, The Mathematics of Paul Erdős I.

[23]  Gábor Tardos,et al.  On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers , 2017, SODA.

[24]  László A. Székely,et al.  Crossing Numbers and Hard Erdős Problems in Discrete Geometry , 1997, Combinatorics, Probability and Computing.

[25]  Nabil H. Mustafa,et al.  Independent set of intersection graphs of convex objects in 2D , 2004, Comput. Geom..

[26]  Jean Cardinal,et al.  Ramsey-type theorems for lines in 3-space , 2016, Discret. Math. Theor. Comput. Sci..

[27]  János Pach,et al.  Applications of a New Separator Theorem for String Graphs , 2013, Combinatorics, Probability and Computing.

[28]  Micha Sharir,et al.  On the Number of Incidences Between Points and Curves , 1998, Combinatorics, Probability and Computing.

[29]  Pavel Valtr,et al.  On Geometric Graphs with No k Pairwise Parallel Edges , 1997, Discret. Comput. Geom..

[30]  J. Kratochvil,et al.  Intersection Graphs of Segments , 1994, J. Comb. Theory, Ser. B.

[31]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

[32]  János Pach,et al.  Coloring kk-free intersection graphs of geometric objects in the plane , 2008, SCG '08.

[33]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1990, Comb..

[34]  Joshua Zahl,et al.  Cutting algebraic curves into pseudo-segments and applications , 2016, J. Comb. Theory A.

[35]  Zeev Dvir,et al.  Incidence Theorems and Their Applications , 2012, Found. Trends Theor. Comput. Sci..

[36]  János Pach,et al.  A Crossing Lemma for Jordan Curves , 2017, Advances in Mathematics.

[37]  William T. Trotter,et al.  Triangle-free intersection graphs of line segments with large chromatic number , 2012, J. Comb. Theory, Ser. B.

[38]  Tamal K. Dey,et al.  Improved Bounds for Planar k -Sets and Related Problems , 1998, Discret. Comput. Geom..

[39]  Rom Pinchasi.  Geometric graphs with no two parallel edges , 2008, Comb..

[40]  E. Szemerédi,et al.  Crossing-Free Subgraphs , 1982 .

[41]  Meir Katchalski,et al.  On Geometric Graphs with No Two Edges in Convex Position , 1998, Discret. Comput. Geom..

[42]  Yakov Shimeon Kupitz.  On Pairs of Disjoint Segments in Convex Position in The Plane , 1984 .

[43]  J. Pach,et al.  Separator theorems and Turán-type results for planar intersection graphs , 2008 .

[44]  Ramsey-type constructions for arrangements of segments , 2010, 1010.2167.

[45]  János Pach,et al.  A Separator Theorem for String Graphs and its Applications , 2009, Combinatorics, Probability and Computing.

[46]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[47]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[48]  Jirí Matousek,et al.  Efficient partition trees , 1992, Discret. Comput. Geom..

[49]  János Pach,et al.  Computing the independence number of intersection graphs , 2011, SODA '11.