PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Combinatorics ↓


Extremal Combinatorics


Extreme combinatorics is a fascinating area of mathematics that focuses on the study of extreme (maximal or minimal) properties of discrete structures. These structures can include graphs, hypergraphs, sets, sequences, and other combinatorial objects. The central questions of extreme combinatorics often revolve around determining the maximum or minimum size of a collection that satisfies given conditions. This can involve determining the largest graph with certain properties or the smallest set that satisfies specific conditions.

Basic concepts and definitions

To dive into extreme combinatorics, we must first understand some basic concepts and definitions that form the foundation of this field.

Graphs and subgraphs

A graph is a collection of vertices (or nodes) and the edges connecting pairs of vertices. A subgraph is a subset of the graph's vertices, plus all the edges connecting them.

Hypergraphs

A hypergraph generalizes the concept of a graph by having edges that can connect more than two vertices. These edges, often called hyperedges, can contain any number of vertices.

Sets and sequences

In combinatorics, sets are collections of distinct objects or numbers, while sequences are ordered collections of objects or numbers. Sets and sequences are commonly used to study different problems, such as maximum intersection or overlapping sequences.

Major theorems in extremal combinatorics

Several major theorems underpin extremal combinatorics. These theorems often provide insights into or bounds on the maximal or minimal properties of combinatorial structures.

Turán's theorem

One of the most famous results in extremal graph theory is Turán's theorem. This theorem provides a bound on the number of edges in a graph that lacks a complete subgraph of a certain size.

Turán's theorem: For a graph with n vertices that does not contain a complete subgraph with r + 1 vertices, the maximum number of edges is:

T_r(n) = left(1 - frac{1}{r} right) frac{n^2}{2}

This implies that if you want a graph that does not include the complete graph on r + 1 vertices, then the number of edges is limited by the above formula.

For example, if n = 5 and r = 2, then the maximum number of edges in a triangle-free graph is:

T_2(5) = left(1 - frac{1}{2} right) frac{5^2}{2} = frac{1}{2} cdot frac{25}{2} = 6.25

Thus, a triangle-free graph with 5 vertices can have at most 6 edges.

Erdős–Stone theorem

The Erdős–Stone theorem generalizes the Turán theorem and is a cornerstone of extremal graph theory, determining the extremal number for any graph with chromatic number greater than 2.

Erdős-Stone Theorem: If a graph G does not have a complete subgraph K_{r+1}, then the maximum number of edges is approximately:

ex(n, K_{r+1}) = left(1 - frac{1}{r} + o(1) right) frac{n^2}{2}
A B C

The above SVG represents a triangle.

Extreme set theory

Extremal set theory generally deals with questions about families of sets. The famous Erdős-Ko-Rado theorem is one of the deepest results in this area.

Erdős–Ko–Rado theorem

This theorem provides an upper bound on the size of a family of sets, if all sets intersect with at least a fixed number of elements.

Erdős-Ko-Rado Theorem: Suppose k ≤ n/2 and F is a family of k- element subsets of {1, 2, ..., n}. If every two sets in F intersect each other, then |F| ≤ C(n-1, k-1). This limit is attained when F consists of all k- element sets containing a fixed element.

For n = 5 and k = 2, the family of sets { {1,2}, {2,3}, {3,4}, {4,5}, {5,1} } attains a limit when all pairs of vertices intersect.

Applications and problems

Extreme combinatorics has applications in various fields, including computer science, information theory, and design theory. It is helpful in solving problems that involve maximizing or minimizing some quantities under given constraints.

Problem 1: Maximizing a triangle-free graph

If a graph has n vertices, what is the maximum number of edges it can have without forming any triangles?

According to Turán's theorem, the answer is:

T_2(n) = left(1 - frac{1}{2} right) frac{n^2}{2} = frac{n^2}{4}

Problem 2: Largest family of disjoint sets

Suppose you have subsets of a set of size n, and you want the largest family of subsets such that no two subsets are pairwise disjoint. The solution to this problem is given by the Erdős-Ko-Rado theorem.

If the subsets are k- element sets, then the largest family is obtained when each set has at least one common element, and the size is at most:

C(n-1, k-1)

Problem 3: Conway's thrackle conjecture

A thrackle is a graph in the plane in which every pair of edges meets exactly once. Conway's thrackle conjecture states that the maximum number of edges in a thrackle is equal to the number of its vertices.

Although this problem has not been solved, it illustrates the intriguing questions posed by extreme combinatorics.

Conclusion

Extreme combinatorics is a robust and vibrant field that poses and resolves fundamental questions about how configurations are maximized and minimized within prescribed structures. It provides important insights not only in pure mathematics but also in computational fields where these principles are applied to solve real-world problems. With its rich history of results and ongoing research, extreme combinatorics remains a major area of mathematical investigation.


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Extremal Combinatorics

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