To a set of n points in the plane, one can associate a graph that has less than n2 vertices and has the property that k-cliques in the graph correspond vertex sets of convex k-gons in the point set. We prove an upper bound of 2k−1 on the size of a planar point set for which the graph has chromatic number k, matching the bound conjectured by Szekeres for the clique number. Constructions of Erdős...

The Strong Dirac conjecture, open in some form since 1951 [5], is that every set of n points in R includes a member incident to at least n/2 − c lines spanned by the set, for some universal constant c. The less frequently stated dual of this conjecture is that every arrangement of n lines includes a line incident to at least n/2 − c vertices of the arrangement. It is known that an arrangement o...

A simplicial arrangement of pseudolines is a collection of topological lines in the projective plane where each region that is formed is triangular. This paper refines and develops David Eppstein’s notion of a kaleidoscope construction for symmetric pseudoline arrangements to construct and analyze several infinite families of simplicial pseudoline arrangements with high degrees of geometric sym...

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n). No known input causes our algorithm to use area Ω(n ) for any > 0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puz...

Given a line arrangement A with n lines, we show that there exists a path of length n/3−O(n) in the dual graph of A formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with 3k blue and 2k red lines with no alternating path longer than 14k. Further, we show that any line arrangement with n lines has a coloring ...

The realizability problem for rank 3 oriented matroids (see [1]) is equivalent to the pseudoline stretchability problem (see [4]). This paper uses an example to illustrate a new approach to this problem. The main theorem of this paper, like the trigonometric form of Ceva’s theorem, shows a non-trivial relationship amongst the angles in a specific line arrangement figure (c). We work in polar co...