Connections between additive combinatorics, graph theory, and incidence geometry

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 \(B\) arecalled mutually avoiding if \(A\) avoids \(B\) and \(B\) avoids \(A .\) Aronov et al. showed that any set of \(n\) points in general positionin 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 productset 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 lowerbounds 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].