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401-3055-64L 5 Credits BSC , DR , MSC D-ITET , D-MATH , D-INFK
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Algebraic Methods in Combinatorics

Lecturers & Examiners: Prof. Dr. Benjamin Sudakov
VVZ CR n/a

Last Updated: 2026-06-01 11:30:51

Abstract

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. This course provides a gentle introduction to Algebraic methods, illustrated by examples and focusing on basic ideas and connections to other areas.

Objective

The students will get an overview of various algebraic methods for solving combinatorial problems. We expect them to understand the proof techniques and to use them autonomously on related problems.

Content

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, well-developed 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 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 Chevalley-Warning theorem. Applications such as: Solution of Kakeya problem in finite fields, counterexample to Borsuk's conjecture, chromatic number of the unit distance graph of Euclidean space, explicit constructions of Ramsey graphs and many others. The course website can be found at https://moodle-app2.let.ethz.ch/course/view.php?id=15757

Resources

Lecture Notes

Lectures will be on the blackboard only, but there will be a set of typeset lecture notes which follow the class closely.

Learning Materials (Links)

General Information

Language
English
Levels
BSC , DR , MSC
Frequency
Every two years

Examination

Type
session examination
Mode
written 180 minutes
Aids
Students are allowed to bring ONLY a printed copy of the lecture notes with no extra writing (highlighting and blank post-its are allowed).

Course Components

Type Title Time & Place Hours
lecture Algebraic Methods in Combinatorics
  • Wed 10:15-12:00 (HG D 5.2)
2 h weekly
exercise Algebraic Methods in Combinatorics
  • Mon 12:15-13:00 (HG E 21)
  • Mon 13:15-14:00 (HG E 21)
1 h weekly

Offered In