Walk through Combinatorics, Fall 2012

Highly non-parallel roads


Mondays, Wednesdays, Fridays 11:30


Wean Hall 8201


If you ever stumbled upon a mathematical object that is finite and is fun to play with, you must have wandered into the land of combinatorics (also known as discrete mathematics). You might have walked there through one of the many well-trodden roads from logic and computer science, or were yanked through some of the recently-discovered wormholes from number theory or topology. Even if you have not been there, you still might want a guide.

In this course we will explore this land, and learn of paths that connect its different parts. We will focus on four areas:

  1. Ramsey theory, where we learn how to find order in the utter chaos. Here the main results are the pigeonhole principle, Ramsey's theorem, theorems of van der Waerden and Hales–Jewett. Much of the time will be devoted to their applications.
  2. Enumerative combinatorics, where we learn how to count not only trees, and forests, but also walks, and cycles. Here we encounter clever bijections, Möbius inversion formula, generating functions, Cauchy's theorem as a tool in asymptotic analysis, as well as a multitude of combinatorial objects to count.
  3. Probabilistic method, where we learn how use the power of randomness to conjure mathematical objects that are both simple and exotic. Our main helpers will be linearity of expectations, second moment method, alteration tricks, Lovász local lemma, and concentration inequalities. We will also see and use the famed Szemerédi regularity lemma.
  4. Arithmetic combinatorics, where we learn how to tame sets of numbers. We will use the sumset inequalities of Plünnecke and Ruzsa, sum-product estimates, as well as theorems of Roth and Balog–Szemerédi–Gowers. If we are lucky, we might even prove Freiman's theorem!


Due to the variety of topics, there is no single book that covers everything. Some good written resources covering parts of the course are

Links to additional resources will be posted as the course progresses.

Student-created resources:

The notes for the class are taken in real-time by JD Nir (text) and Will Macrae (pictures). The notes are like a silent film — they are visually great, but lack the voiceovers. Use them, but refer to the written resources above and to myself if something is unclear.

More fun:

More fun can be had at my office hours on Wednesdays 2:30–3:30pm in Wean 6202 or 6th floor lounge. I am also available by appointment.

Course activities:

Mastery of combinatorics requires practice. Hence, there will be regular homeworks. You are strongly encouraged to do homework individually. Collaboration and use of external sources are permitted, but discouraged, and must be fully acknowledged and cited. Collaboration may involve only discussion; all the writing must be done individually. The homeworks will be returned one week after they are due.

Students are expected to fully participate in the class. Discussions during the lectures are encouraged.

The final grade will be a monotone nondecreasing function of the total homework grade. There will be opportunities for extra credit. Unusually insightful solutions, and other achievements will be appropriately rewarded.


Back to the homepage