This thesis includes several results on the extremal combinatorics of
hyperplane arrangements, kinetic point sets, and other geometric struc-
tures. It consists of three papers published or submitted elsewhere. “A
Bichromatic Incidence Bound and an Application”, joint with George
Purdy and Justin Smith, was published in Discrete and Computational
Geometry. I presented “Collinearities in Kinetic Point Sets”, joint with
George Purdy, Justin Smith, and Csaba T¿¿¿¿th, at the 23rd Canadian Con-
o
ference on Computational Geometry, 2011. “Some Results Related to a
Conjecture of Dirac’s”, joint with George Purdy and Justin Smith, has
been submitted to The Electronic Journal of Combinatorics.