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.Let V be a set of n points in general position in the plane, that is, assume that no 3 points of V are collinear. A geometric graph is a graph G = (V, E) whose vertex set is V and whose edges are represented by possibly crossing straight-line segments connecting certain pairs of points in V . If every pair of points in V is connected by a segment, we have E = V 2 , and G is called a complete geometric graph. Two edges pq, p q ∈ E are said to cross if the corresponding segments share an interior point. Topological graphs are defined similarly, except that their edges can be represented by any Jordan curves that have no interior points that belong to V .Crossing patterns and intersection graphs. Finding maximum cliques or independent sets in intersection graphs of segments, rays, and other convex sets in the plane is a computationally hard problem and a classic topic in computational and combinatorial geometry [4,10,21,30,31,32]. There are many interesting Ramsey-type problems and results about the existence of large cliques or large independent sets in intersection graphs of segments [7,11,34,35,42] and, more generally, of Jordan curves ("strings") [20,24,22]. Some of these questions are intimately related to counting incidences between points and lines [41,45,46], and to bounding the complexity of k-levels in arrangements of lines in R 2 [16].It appears to be a somewhat simpler task to understand the combinatorial structure of crossings between the edges of a geometric or topological graph. Despite decades of steady progress, we have very few asymptotically tight results in this direction. Perhaps the best known and most applicable theorem of this kind is the so-called Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi [6] and Leighton [36], which states that any topological graph G = (V, E) with |E| > 4|V | determines at least Ω |E| 3 /|V | 2 crossing pairs of edges. Recently, a similar result has been established by the authors for contact graphs of families of Jordan curves [40].According to another asymptotically tight result, for t > 1, every geometric graph G with |E| ≥ nt edges has two disjoint sets of edges, E 1 , E 2 ⊂ E, each of size Ω(t), such that every edge in E 1 crosses all edges in E 2 ; see, e.g., [20, Theorem 6]. A similar theorem holds for topological graphs, with the difference that then |E 1 |, |E 2 | = Ω (t/ log t) [22]. It is a major unsolved question to decide whether under these circumstances G must also contain a family of pairwise crossing edges, whose size tends to infinity as t → ∞. It is conjectured that one can always choose such a family consisting of almost t edges. If this stronger conjecture is true for t 1−ot(1) edges, then for t ≈ n/2, it would ...

A positive fraction Erdos-Szekeres theorem and its applications

“…This corollary is also asymptotically best possible. To see this, for n > (k − 1) 2 > 0 and a sequence A = (a i ) n i=1 of distinct real numbers, we can apply Corollary 6.1 with = (64k) −1 to A and obtain an -monotone subsequence S and then apply Lemma 2.1 in [13] to S to conclude a blockmonotone subsequence of depth k and block-size Ω(n/k 2 ). So Corollary 6.1 implies Theorem 1.1.…”

“…Finally, let us remark that a recent result due to Pach, Rubin, and Tardos [13] shows that every n-element planar point set in general position determines at least n/e O( √ log n) pairwise crossing segments. By using Theorem 1.3 instead of Lemma 3.3 from their paper, one can improve the constant hidden in the O-notation.…”

On the structure of pointsets with many collinear triples

“…Given three pointsets in the plane, P, Q, and S, we say that the three sets are mutually avoiding set if no line determined by two points in a set intersects the two convex hulls of the other two pointsets. The concept of mutually avoiding sets are often used in discrete and computational geometry, like in [2], [40], [25] and [27]. It was proved in [2] that any planar n-element pointset in general position contains two mutually avoiding subsets of size at least n/12.…”

Perfect Matchings with Crossings

“…Given n, perfect matching with exactly µ = n/2 2 crossings belongs to an active area of research around so-called crossing-family (see for example [8,9]). Only very recently it was shown that any set of n points in general position contains a crossing families (a perfect matching with µ crossings) of almost linear size, more precisely, of size n 1−o(1) [10]. This was the first substantial improvement after 25 years.…”