A Note on Turán Numbers and the Erdős-Stone-Simonovits Theorem

Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Turán number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Turán number of graphs is a central topic in extremal graph theory. The purpose of this article is to present some well-known results about this field but also to prove the Erdős-Stone-Simonovits theorem in an original manner.