On mutually avoiding sets

Two finite sets of points in the plane are called mutually avoiding if any straight line passing through two points of anyone of these two sets does not intersect the convex hull of the other set. For any integer n, we construct a set of n points in general position in the plane which contains no pair of mutually avoiding sets of size more than O (n). The given bound is tight up to a constant factor, since Aronov et al. [AEGKKPS] showed a polynomial time algorithm for finding two mutually avoiding sets of size (n) in any set of n points in general position in the plane

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are n^{1/2} lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are (n/3)^{1/2} of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs

Eigenvalue Etch-A-Sketch

Paul Erdo ̋s’s Empty Hexagon Problem asks if there exists a number H(6) such that for all sets of n ≥ H points in general position on the plane six of the points form the vertices of an empty convex hexagon. This problem is open

On the structure of pointsets with many collinear triples

It is conjectured that if a finite set of points in the plane contains many collinear triples then there is some structure in the set. We are going to show that under some combinatorial conditions such pointsets contain special configurations of triples, proving a case of Elekes' conjecture. Using the techniques applied in the proof we show a density version of Jamison's theorem. If the number of distinct directions between many pairs of points of a pointset in convex position is small, then many points are on a conic

On Distinct Angles in the Plane

We prove that if $N$ points lie in convex position in the plane then they determine $N^{1+ 3/23-o(1)}$ distinct angles, provided the points do not lie on a common circle. This is the first super-linear bound on the distinct angles problem that has received recent attention. This is derived from a more general claim that if $N$ points in the convex position in the real plane determine $KN$ distinct angles, then $K=\Omega(N^{1/4})$ or $\Omega(N/K)$ points are co-circular. The proof makes use of the implicit order one can give to points in convex position, recently used by Solymosi. Convex position also allows us to divide our set into two large ordered pieces. We use the squeezing lemma and the level sets of repeated angles to estimate the number of angles formed between these pieces. We obtain the main theorem using Stevens and Warren's convexity versus sum-set bound.Comment: 23 pages, 10 figure