This research project examines several important problems in Ramsey theory and discrete geometry. These are problems arising from both combinatorics and geometry, with the common theme of determining how large or small a certain finite set can be under certain restrictions. For example, given a set of n points in the plane, how often can the unit distance occur among them? Many of these problems have fascinated mathematicians for decades, while others are fueled by the more recent development of computational geometry. The PI will tackle these problems using a wide range of mathematical tools and techniques from probability, topology, algebraic geometry, and combinatorics. One of the main goals of this project is to study the interplay between combinatorial and algebraic methods, and to further develop these methods by studying some of the most central open problems in the field. The PI will also continue to encourage high school, undergraduate, and graduate students to work in combinatorial research, and continue to teach courses which cover the latest results and methods used in combinatorics and discrete geometry.
The first area in this project examines Ramsey-type problems in combinatorics and geometry. One of the major goals of this project is to obtain new bounds for classical hypergraph Ramsey numbers. The PI will also continue a sequence of recent works on Ramsey-type problems for hypergraphs arising in geometry, which include the Erdos-Szekeres convex polygon problem in higher dimensions, and finding large independent sets in intersection graphs of geometric objects. The second area of this project explores various extensions of the famous Szemeredi-Trotter theorem, and its applications in additive number theory. Specific problems include characterizing dense point-line arrangements, and estimating the number of incidences between points and curves in the plane.