5 research outputs found
A Crossing Lemma for Jordan curves
If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a touching point. The main result of this paper is a Crossing Lemma for simple curves: Let X and T stand for the sets of intersection points and touching points, respectively, in a family of n simple curves in the plane, no three of which pass through the same point. If |T|>cn, for some fixed constant c>0, then we prove that |X|=Ω(|T|(logâĄlogâĄ(|T|/n))1/504). In particular, if |T|/nââ then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between n pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least (1âo(1))n2. © 201
The number of tangencies between two families of curves
We prove that the number of tangencies between the members of two families,
each of which consists of pairwise disjoint curves, can be as large as
. We show that from a conjecture about forbidden -
matrices it would follow that this bound is sharp for doubly-grounded families.
We also show that if the curves are required to be -monotone, then the
maximum number of tangencies is , which improves a result by
Pach, Suk, and Treml. Finally, we also improve the best known bound on the
number of tangencies between the members of a family of at most
-intersecting curves
Planar Point Sets Determine Many Pairwise Crossing Segments
We show that any set of points in general position in the plane
determines pairwise crossing segments. The best previously known
lower bound, , was proved more than 25 years ago by
Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof
is fully constructive, and extends to dense geometric graphs.Comment: A preliminary version to appear in the proceedings of STOC 201