On the Richter–Thomassen Conjecture about Pairwise Intersecting Closed Curves †

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1 - o(1)) n(2). We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R-2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Omega(nt root logt/log log t).

[1]  Timothy M. Chan.  On Levels in Arrangements of Curves, II: A Simple Inequality and Its Consequences , 2005, Discret. Comput. Geom..

[2]  Timothy M. Chan.  On levels in arrangements of curves, iii: further improvements , 2008, SCG '08.

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

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

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

[6]  Hisao Tamaki,et al.  How to Cut Pseudoparabolas into Segments , 1998, Discret. Comput. Geom..

[7]  József Solymosi,et al.  An Incidence Theorem in Higher Dimensions , 2012, Discret. Comput. Geom..

[8]  Timothy M. Chan.  On Levels in Arrangements of Curves , 2003, Discret. Comput. Geom..

[9]  Micha Sharir,et al.  On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles , 1986, Discret. Comput. Geom..

[10]  Gelasio Salazar,et al.  On the Intersections of Systems of Curves , 1999, J. Comb. Theory, Ser. B.

[11]  Dhruv Mubayi.  Intersecting Curves in the Plane , 2002, Graphs Comb..

[12]  H. Halkin,et al.  Discretional Convexity and the Maximum Principle for Discrete Systems , 1966 .

[13]  Micha Sharir,et al.  Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces , 2002, Discret. Comput. Geom..

[14]  Endre Szemerédi,et al.  A Combinatorial Distinction Between the Euclidean and Projective Planes , 1983, Eur. J. Comb..

[15]  János Pach,et al.  Beyond the Richter-Thomassen Conjecture , 2016, SODA.

[16]  Micha Sharir,et al.  Pseudo-line arrangements: duality, algorithms, and applications , 2002, SODA '02.

[17]  Micha Sharir,et al.  Lenses in arrangements of pseudo-circles and their applications , 2004, JACM.

[18]  Carsten Thomassen,et al.  Intersections of curve systems and the crossing number ofC5×C5 , 1995, Discret. Comput. Geom..