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

TL;DR

Problem: determine $ex(n,H)$ for fixed $H$ and its dependence on $\chi(H)$. The authors present an elementary, self-contained approach to the Erdos-Stone-Simonovits theorem that avoids Szemeredi's lemma and revisits classical results (Mantel, Turan, Kovari-Sos-Turan, Bondy-Simonovits) with concise proofs. Key contributions include bounds for cliques, bipartite graphs, and even cycles, culminating in the asymptotic $ex(n,G) \le \frac{n^2}{2}\left(1-\frac{1}{\chi(G)-1}+\epsilon\right)$ for large $n$. Methods rely on elementary counting, convexity, BFS layering, and embedding arguments to illuminate the density–chromatic-number tradeoff in Turán-type problems.

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 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.

Theorems & Definitions (21)

• Theorem 2.1: Mantelmantel1907
• proof : First proof of Theorem \ref{['thm-Mantel']}
• proof : Second proof of Theorem \ref{['thm-Mantel']}
• Theorem 2.2: Turánturan1941
• proof : Proof of Theorem \ref{['thm- Turán']}
• Theorem 3.1: Kóvári-Sós-Turánkovari1954
• proof : Proof of Theorem \ref{['thm-Kovari-Sos-Turan']}
• Remark 3.2
• Theorem 4.1: Erdőserdos1964
• proof : Proof of the Theorem \ref{['thm-Erdős']}