UC San Diego

One of the Erd\H{o}s-like cornerstones in incidence geometry from which many other results follow is the celebrated Szemer edi-Trotter Theorem which states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this thesis, we study the effect of forbidding grids and short even cycles on the incidence graphs of point-line arrangements in the plane.

Let \(A\) and \(B\) be two disjoint finite sets of points in the plane such that their union contains no three points on a line. We say that \(A\) \emph{avoids} \(B\) if no straight line determined by a pair of points in \(A\) intersects the convex hull of $B.$ $A$ and \(B\) are

called mutually avoiding if \(A\) avoids \(B\) and \(B\) avoids \(A .\) Aronov et al. showed that any set of \(n\) points in general position

in the plane contains a pair of mutually avoiding sets, each of size at least \(\Omega(\sqrt{n})\). Moreover, they proved that any set of \(n\) points in general position in \(\mathbb{R}^{d}\) contains a pair of mutually avoiding sets, each of size at least \(\Omega\left(n^{\frac{1}{d^{2}-d+1}}\right)\). In this thesis, we give a generalized version of mutually avoiding set theorem in the plane.

Given an algebraic structure \(R\) and a subset \(A \subset R,\) define the sum set and the product

set of \(A\) to be \(A+A=\{a+b: a, b \in A\}\) and \(A \cdot A=\{a \cdot b: a, b \in A\}\) respectively.

Showing under what conditions at least one of \(|A+A|\) or \(|A \cdot A|\) is large has a long history of study that continues to the present day. By employing recent developments on the energy of polynomials over finite fields, we give the best-known lower

bounds on $\max\{|A+A|, |f(A,A)|\}$, when $A$ is a small subset of $ \mathbb{F}_p,$ and $f$ is a quadratic non-degenerate polynomial in $\mathbb{F}_p[x,y].$