A Note on Tur\'an Numbers and the Erd\H{o}s-Stone-Simonovits Theorem

Abstract

Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Tur\'an number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Tur\'an 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 Erd\H{o}s-Stone-Simonovits theorem in an original manner.