🎲Extremal Combinatorics Unit 3 – Turán's Theorem: Extremal Graph Theory

Key Concepts and Definitions

• Extremal graph theory studies the maximum or minimum values of graph parameters under certain constraints
• Turán's theorem provides an upper bound on the number of edges in a graph that does not contain a specific subgraph
• The Turán graph $T(n,r)$ is a complete multipartite graph with $n$ vertices and $r$ partite sets of nearly equal size
  • For example, $T(6,2)$ is a complete bipartite graph with partite sets of size 3 and 3
• The Turán number $ex(n,F)$ represents the maximum number of edges in an $n$-vertex graph that does not contain $F$ as a subgraph
• The Turán density $\pi(F)$ of a graph $F$ is defined as $\lim_{n\to\infty} \frac{ex(n,F)}{\binom{n}{2}}$
• A graph is $F$-free if it does not contain $F$ as a subgraph
• The chromatic number $\chi(G)$ of a graph $G$ is the minimum number of colors needed to color its vertices so that no two adjacent vertices have the same color

Historical Context and Development

• Turán's theorem was proved by Hungarian mathematician Pál Turán in 1941
• The theorem was motivated by a question posed by Turán's friend, Pál Erdős, about the maximum number of edges in a triangle-free graph
• Turán's original proof used a counting argument and the pigeonhole principle
• Later proofs of Turán's theorem employed various techniques, including induction, the regularity lemma, and the probabilistic method
• The development of Turán's theorem led to the growth of extremal graph theory as a subfield of combinatorics
• Turán's theorem has been generalized to other forbidden subgraphs and hypergraphs
• The study of Turán-type problems has inspired numerous related questions and conjectures in extremal combinatorics

Statement of Turán's Theorem

• Turán's theorem states that for any integer $r \geq 2$ and $n \geq 1$, the Turán graph $T(n,r)$ is the unique $n$-vertex $K_{r+1}$-free graph with the maximum number of edges
  • In other words, any $n$-vertex graph with more edges than $T(n,r)$ must contain a copy of $K_{r+1}$
• The number of edges in the Turán graph $T(n,r)$ is given by $\left\lfloor \frac{r-1}{2r}n^2 \right\rfloor$
• Equivalently, Turán's theorem can be stated in terms of the Turán number: $ex(n,K_{r+1}) = \left\lfloor \frac{r-1}{2r}n^2 \right\rfloor$
• The theorem provides a sharp upper bound on the number of edges in a $K_{r+1}$-free graph
• Turán's theorem can be generalized to other forbidden subgraphs $F$, giving an upper bound on $ex(n,F)$

Proof Techniques and Approaches

• Turán's original proof used a counting argument and the pigeonhole principle
  • The proof shows that any $n$-vertex graph with more edges than $T(n,r)$ must contain a vertex with degree at least $\frac{r}{r+1}n$, which implies the existence of a $K_{r+1}$
• An alternative proof uses induction on $n$ and the concept of vertex deletion
  • The inductive step involves removing a vertex of maximum degree and applying the inductive hypothesis to the remaining graph
• The regularity lemma, a powerful tool in graph theory, can be used to prove a strengthened version of Turán's theorem
  • The regularity lemma allows for the decomposition of a large graph into a bounded number of pseudo-random bipartite subgraphs
• The probabilistic method, which employs random constructions and probabilistic arguments, has been used to prove lower bounds on Turán numbers
• Other proof techniques for Turán-type problems include the stability method, the absorption method, and the container method

Applications and Examples

• Turán's theorem has applications in various areas, including computer science, coding theory, and social network analysis
• In computer science, Turán's theorem is used in the analysis of graph algorithms and the design of efficient data structures
  • For example, the theorem can be applied to bound the complexity of certain graph coloring algorithms
• Turán's theorem is closely related to Ramsey theory, which studies the emergence of ordered structures in large random sets
  • Ramsey numbers can be used to derive lower bounds on Turán numbers
• In social network analysis, Turán's theorem can be used to study the structure of social graphs and the formation of cliques
  • The theorem provides insights into the maximum number of connections in a network without creating tightly connected subgroups
• Examples of Turán numbers for specific graphs:
  • $ex(n,K_3) = \left\lfloor \frac{n^2}{4} \right\rfloor$ (triangle-free graphs)
  • $ex(n,C_4) = \frac{1}{2}n^{3/2} + o(n^{3/2})$ (graphs without 4-cycles)
  • $ex(n,K_{2,2}) = \frac{1}{2}n^{3/2} + o(n^{3/2})$ (graphs without 4-cycles)

Related Theorems and Extensions

• The Erdős-Stone-Simonovits theorem is a generalization of Turán's theorem that provides an asymptotic estimate for the Turán number of any non-complete graph
  • The theorem states that $ex(n,F) = \left(1 - \frac{1}{\chi(F)-1}\right)\binom{n}{2} + o(n^2)$ for any non-complete graph $F$
• The Kővári-Sós-Turán theorem is an extension of Turán's theorem to bipartite graphs
  • The theorem provides an upper bound on the number of edges in a bipartite graph that does not contain a specific complete bipartite subgraph
• The Erdős-Rényi random graph model, denoted by $G(n,p)$, is closely related to Turán's theorem
  • The threshold for the emergence of a specific subgraph in $G(n,p)$ is related to the Turán density of that subgraph
• The Erdős-Simonovits stability theorem characterizes the structure of graphs that are close to the Turán number
  • The theorem states that graphs with slightly fewer edges than the Turán number must be structurally similar to the Turán graph
• Hypergraph Turán problems investigate the maximum number of edges in a hypergraph that does not contain a specific subhypergraph
  • Turán-type results for hypergraphs are generally more challenging and less well-understood than their graph counterparts

Problem-Solving Strategies

• When approaching a Turán-type problem, first identify the forbidden subgraph or subgraphs
• Consider the structure of the Turán graph for the given forbidden subgraph
  • The Turán graph provides an upper bound on the number of edges and can guide the construction of extremal examples
• Analyze the properties of the forbidden subgraph, such as its chromatic number, to derive bounds on the Turán number
• Use known Turán numbers for specific graphs as building blocks for more complex problems
  • For example, the Turán numbers for complete graphs and complete bipartite graphs can be used to derive bounds for other subgraphs
• Apply proof techniques such as induction, the regularity lemma, or the probabilistic method, depending on the nature of the problem
• Consider the asymptotic behavior of the Turán number and use asymptotic notation to simplify expressions
• Look for connections to related problems in extremal graph theory, such as Ramsey numbers or the Erdős-Stone-Simonovits theorem
• Utilize symmetry and structural properties of the graphs involved to simplify the problem or reduce it to a known case

Further Reading and Resources

• "Extremal Graph Theory" by Béla Bollobás is a comprehensive textbook that covers Turán's theorem and related topics in depth
• "Modern Graph Theory" by Béla Bollobás includes a chapter on extremal graph theory and Turán-type problems
• "Extremal Combinatorics" by Stasys Jukna provides a broad overview of extremal problems in combinatorics, including graph theory
• The survey paper "An Invitation to Extremal Graph Theory" by Zoltán Füredi and Miklós Simonovits offers an accessible introduction to the field
• The "Handbook of Graph Theory" edited by Jonathan L. Gross, Jay Yellen, and Ping Zhang contains several chapters on extremal graph theory and Turán-type problems
• The online database "The On-Line Encyclopedia of Integer Sequences" (OEIS) contains a wealth of information on Turán numbers and related sequences
• The "Extremal Combinatorics" course notes by Benny Sudakov provide a concise introduction to the topic and include examples and exercises
• The "Extremal Graph Theory" course notes by Jacques Verstraëte cover Turán's theorem and related topics in detail, with an emphasis on recent developments in the field