
Lecturer: Prof. Benny Sudakov
Thursday 13:0015:00, HG F 3
Assistant: Dániel Korándi
Tuesday 11:0012:00, CAB G 52
Course Description:
Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage and often relies on deep, welldeveloped tools.
One of the main general techniques that played a crucial role in the development of Combinatorics was the application of algebraic methods. 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. 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.
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 ChevalleyWarning 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.
Assignments:
Course Syllabus:
We will cover the following topics this semester. This outline may be updated as the term progresses.
Introduction 
Rules and Clubs Lindström's theorem Joints Points in the Euclidean space with only two distances Kakeya problem in finite fields Consistent edgecolorings of the complete graph Number of lines determined by noncollinear points in the plane Nonuniform Fisher's inequality 
Set systems with restricted intersections: 
RayChaudhuri–Wilson theorem Frankl–Wilson theorem 
Applications of intersection theorems 
Explicit constructions of Ramsey graphs Chromatic number of the unit distance graph of the Euclidean space Counterexample to Borsuk's conjecture 
Convexity 
Radon lemma and Helly's theorem Existence of Centerpoint Colorful Carathéodory theorem Tverberg's theorem 
Eigenvalues 
definition, simple properties, computation for few basic families of graphs Applications: Decomposition of K_10 into Petersen's graphs Variational definition of eigenvalues and applications to interlacing, bounding independence number, chromatic number, max cut Expansion and spectral gap Bounding matrix ranks and Johnson–Lindenstrauss lemma 
Chevalley–Warning theorem and its applications 
3regular subgraphs of 4regular graphs Davenport constants of abelian groups Blocking sets in affine hyperplanes Zero sum sets and the Erdős–Ginzburg–Ziv lemma 
Combinatorial Nullstellensatz, Bounds on the size of the sum of two subsets in Z_p 
Cauchy–Davenport theorem and Erdős–Heilbronn conjecture Covering cubes by hyperplanes List chromatic number, applications of the Combinatorial Nullstellensatz to graph colorings 
Wichtiger Hinweis:
Diese Website wird in älteren Versionen von Netscape ohne
graphische Elemente dargestellt. Die Funktionalität der
Website ist aber trotzdem gewährleistet. Wenn Sie diese
Website regelmässig benutzen, empfehlen wir Ihnen, auf
Ihrem Computer einen aktuellen Browser zu installieren. 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