Problems in Discrete Geometry, Incidence Geometry,and Extremal Graph Theory

In this thesis, we study several problems from discrete geometry, incidence geometry, and extremal graph theory. In Chapter 1, we discuss some results in discrete geometry. We study three different but similar discrete geometry problems, which share a similar idea on constructions. The problems ask for sets of points in d-dimensional space that are evenly distributed, and the notion of being evenly distributed is defined in different ways. In Chapter 2, we show sharp results on two problems in incidence geometry: the finite field Kakeya problem and the joints problem. The correct orders of magnitude for these two problems were both obtained by the dimension counting polynomial method. We improved both results by considering carefully chosen spaces of polynomials. The sharp bound on the joints problem also provides a new proof of the Kruskal-Katona theorem and its generalization. We will discuss this connection in this chapter and in chapter 3. In Chapter 3, we study several extremal problems on (hyper)graphs. In the first part of this chapter, we proved some of the Kruskal-Katona-type problems via the entropy method, including a short and new proof of Lov ́asz's version of the Kruskal-Katona theorem. In the second part of this chapter, we give a simplified version of Chase's proof of the Gan-Loh-Sudakov conjecture