Combinatorial geometry of point sets with collinearities

In this thesis we study various combinatorial problems relating to the geometry of point sets in the Euclidean plane. The unifying theme is that all the problems involve point sets that are not in general position, but have some collinearities. As well as giving rise to natural and interesting problems, the study of point sets with collinearities has important connections to other areas of mathematics such as number theory. Dirac conjectured that every set P of n non-collinear points in the plane contains a point in at least n2 − c lines determined by P , for some constant c. It is known that some point is in Ω(n) lines determined by P . We show that some point is in at least n 37 lines determined by P . Erdős posed the problem to determine the maximum integer f(n, `) such that every set of n points in the plane with at most ` collinear contains a subset of f(n, `) points with no three collinear. First we prove that if ` 6 O( √ n) then f(n, `) > Ω( √ n/ ln `). Second we prove that if ` 6 O(n(1− )/2) then f(n, `) > Ω( √ n log` n), which implies all previously known lower bounds on f(n, `) and improves them when ` is not constant. Our results answer a symmetric version of the problem posed by Gowers, namely how many points are required to ensure there are q collinear points or q points in general position. The visibility graph of a finite set of points in the plane has an edge between two points if the line segment between them contains no other points. We establish bounds on the edgeand vertex-connectivity of visibility graphs. We find that every minimum edge cut is the set of edges incident to a vertex of minimum degree. For vertex-connectivity, we prove that every visibility graph with n vertices and at most ` collinear vertices has connectivity at least n−1 `−1 , which is tight. We also prove that the vertex-connectivity is at least half the minimum degree. We study some questions related to bichromatic point sets in the plane. Given two disjoint point sets A and B in the plane, the bivisibility graph