Here we overview some of the methods and results of extremal graph and hypergraph theory. A few geometric applications are also given.

In this thesis we study a variety of problems in combinatorial and computational geometry, which deal with various aspects of arrangements of geometric objects, in the plane and in higher dimensions. Some of these problems have algorithmic applications, while others provide combinatorial bounds for various structures in such arrangements. The thesis involves two main themes: (i) Counting Crossi...

iii 1 The Erdős Distance Problem 1 2 Incidence Geometry 5 2.1 The distinct distances incidence problem . . . . . . . . . . . . . . . . . 5 2.2 The geometry of Elekes’ incidence problem . . . . . . . . . . . . . . . . 11 2.3 Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Dvir’s Polynomial Method 17 3.1 The Polynomial Method . . . . . . . . . . . . . . . . . . ....

Many geometries over fields have formal analogues which can be thought of as geometries over the field of 1 element1. For example, the projective plane over the field Fq has q2 + q+ 1 points and the same number of lines; every line in the plane has q + 1 points. When q = 1, we have a plane with three points and three lines, i.e. a triangle. The flag complex of the triangle is a thin building of...

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