### Tools and fundamental constructions of combinatorial mathematics

The most fruitful such tool is the dimension argument.

Roughly speaking, the method can be described as follows. In order to bound the cardinality of a discrete structure A, one maps its elements to vectors in a linear space, and shows that the set A is mapped to linearly independent vectors.

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It then follows that the cardinality of A is bounded by the dimension of the corresponding linear space. This simple idea is surprisingly powerful and has many famous applications.

4. Counting

This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas. The topics covered in the class will include but are not limited to :. Basic dimension arguments, Spaces of polynomials and tensor product methods, Eigenvalues of graphs and their application, the Combinatorial Nullstellensatz and the Chevalley-Warning theorem. Applications such as: Solution of the Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of the Euclidean space, explicit constructions of Ramsey graphs, and many others.

We will cover the following topics this semester. This outline may be updated as the term progresses. Weitere Informationen finden Sie auf folgender Seite. Important Note: The content in this site is accessible to any browser or Internet device, however, some graphics will display correctly only in the newer versions of Netscape. To get the most out of our site we suggest you upgrade to a newer browser. More information. Lecture homepages old Student Seminars old. Lecture Homepages. Herbstsemester Algebra I.

Algebraic Topology I. Algebraic Methods in Combinatorics. Analysis I. Calculus of Variations. Commutative Algebra. List structures include the incidence list , an array of pairs of vertices, and the adjacency list , which separately lists the neighbors of each vertex: Much like the incidence list, each vertex has a list of which vertices it is adjacent to. Matrix structures include the incidence matrix , a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix , in which both the rows and columns are indexed by vertices.

In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph.

The distance matrix , like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a shortest path between two vertices. There is a large literature on graphical enumeration : the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer A common problem, called the subgraph isomorphism problem , is finding a fixed graph as a subgraph in a given graph.

One reason to be interested in such a question is that many graph properties are hereditary for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem. For example:. One special case of subgraph isomorphism is the graph isomorphism problem. It asks whether two graphs are isomorphic. It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time.

A similar problem is finding induced subgraphs in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A minor or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some or no edges. Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too.

For example, Wagner's Theorem states:. A similar problem, the subdivision containment problem, is to find a fixed graph as a subdivision of a given graph. A subdivision or homeomorphism of a graph is any graph obtained by subdividing some or no edges. Subdivision containment is related to graph properties such as planarity.

For example, Kuratowski's Theorem states:. Another problem in subdivision containment is the Kelmans—Seymour conjecture :. Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their point-deleted subgraphs. Many problems and theorems in graph theory have to do with various ways of coloring graphs.

Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges possibly so that no two coincident edges are the same color , or other variations. Among the famous results and conjectures concerning graph coloring are the following:. Constraint modeling theories concern families of directed graphs related by a partial order.

In these applications, graphs are ordered by specificity, meaning that more constrained graphs—which are more specific and thus contain a greater amount of information—are subsumed by those that are more general.

### by Heinz Luneburg

Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. The unification of two argument graphs is defined as the most general graph or the computation thereof that is consistent with i. For constraint frameworks which are strictly compositional , graph unification is the sufficient satisfiability and combination function.

Well-known applications include automatic theorem proving and modeling the elaboration of linguistic structure. There are numerous problems arising especially from applications that have to do with various notions of flows in networks , for example:. Covering problems in graphs are specific instances of subgraph-finding problems, and they tend to be closely related to the clique problem or the independent set problem. Decomposition, defined as partitioning the edge set of a graph with as many vertices as necessary accompanying the edges of each part of the partition , has a wide variety of question.

Often, it is required to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles. Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:. From Wikipedia, the free encyclopedia.

## Combinatorial Constructions in Ergodic Theory and Dynamics

This article is about sets of vertices connected by edges. For graphs of mathematical functions, see Graph of a function. For other uses, see Graph disambiguation. Area of discrete mathematics. Main article: Directed graph. Main article: Graph drawing. Main article: Graph abstract data type. Main article: Graph coloring. Whitney, Hassler. European Physical Journal B. Bibcode : EPJB Brain Imaging and Behavior. Relativistic Quantum Fields. New York: McGraw-Hill. Journal of Applied Physics. Bibcode : JAP Redesigned network strictly based on Moreno , Who Shall Survive. Discrete mathematics and its applications 7th ed.

Bibcode : Natur.. Freeman and Company, p. Mannheim: Bibliographisches Institut Part I. Discharging", Illinois J. Part II. Reducibility", Illinois J. Areas of mathematics. Category theory Information theory Mathematical logic Philosophy of mathematics Set theory. Abstract Elementary Linear Multilinear. Calculus Real analysis Complex analysis Differential equations Functional analysis.

## Algebra, Combinatorics and Number Theory - Department of Mathematics and Statistics

Combinatorics Graph theory Order theory Game theory. Arithmetic Algebraic number theory Analytic number theory Diophantine geometry. Algebraic Differential Geometric.