Instructor: Jozsef Solymosi

Office: MATH 220

solymosi@math.ubc.ca

Prerequisites: Familiarity with discrete structures (such as MATH 443) is desirable but not imperative. Talented undergraduate students are welcome however good grades and some honours math courses are required.

Tue Thu 11:00 - 12:30 (no class on Tuesday, Sep. 6) Mathematics Annex 1118

Grading: Homework assignments 40%, Take home midterm 20%, Take home final 40%.

Course book (recommended):

Laszlo Lovasz, Combinatorial Problems and Exercises

There is another nice book which is a great read; it is Stasys Jukna’s Extremal Combinatorics. (draft)

Course Material and Topics:

Discrete mathematics (combinatorics) is a fundamental mathematical discipline as well as essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. While in the past many results in this area 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.

The main aim of this course is to introduce many core ideas from this topic, such as extremal set theory, Ramsey theory and design theory as well as some of the dominant proof techniques such as probabilistic methods and algebraic methods.

The course outline follows (with sections taken from Lovasz’ book) with approximately 1 to 2 lectures being spent on each topic.

. Basic enumeration methods (section 1)

. Connectivity in Graphs and Menger’s theorem (section 6)

. Factors of Graphs and Tutte’s theorem (section 7)

. Graph colourings and perfect graphs (section 9)

. Block designs and Wilson’s theorem

. Extremal graph theory and Turan’s theorem (section 10)

. Strongly regular graphs and algebraic methods (section 11)

. Hypergraphs and extremal set theory (section 13)

. Ramsey Theory and probabilistic methods (section 14)

Further Notes (Check the course books above for the details):

Problem sets:

- Enumerations. (Due Sept 22.) Solutions (thanks to Robert)
- Counting. (Due Oct 6.)
- Probability. (Due Oct 18.)
- Midterm
- Ramsey Theory. (Due Nov 17.)