The Szemerédi Regularity Lemma

The Szemerédi Regularity Lemma is a deep result in graph theory which roughly states that large, dense graphs can be approximated by random graphs. The lemma is most helpful in proofs where it may be hard to prove a result for a large graph but could be proven for a smaller random graph. This paper gives an overview of the lemma including relevant definitions and the proof of the theorem. The main importance of the theorem can be found in applications in several disciplines of mathematics such as extremal graph theory, Ramsey theory, and number theory. The main focus of the paper is to demonstrate the use of the lemma in several applications including the Triangle Removal Lemma, Roth's Theorem, the Erdős-Stone theorem and more.