Extremal problems in combinatorics and geometry

This thesis is comprised of four chapters relating to combinatorics and geometry. More specifically, the main topics of the dissertation are incidence geometry and Euclidean Ramsey theory. In Chapter 2, we study the Erdős unit distance problem. In particular, we prove a structural result for pointsets determining many unit distances from a small direction set in any norm. Then, we use this to deduce that, in the Euclidean norm, such pointsets cannot exist. In chapter 3, we give new sharp constructions for the Szemerédi-Trotter problem, where the coordinates of the points come from arbitrary number fields. This adds to previous constructions of Erdős, Elekes, Sheffer and Silier, and Guth and Silier. In Chapter 4, we study a problem in Euclidean Ramsey theory. In particular, we show that in any two-coloring of the plane and positive real number α, there exists a monochromatic three-term arithmetic progression with common distance α. This also proves a significant unsolved case in a conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus. In Chapter 5, we show some negative results in Euclidean Ramsey Theory. In particular, we give red/blue colorings of n-dimensional Euclidean space that avoid short arithmetic progressions in red and long ones in blue. This continues a recent line of work of Conlon and Wu and Führer and Tóth.