### Walk through Combinatorics, Fall 2013

Highly non-parallel roads

#### When:

Mondays, Wednesdays, Fridays 9:30#### Where:

Wean Hall 7201#### What:

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:

*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.*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.*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.*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!

#### Resources:

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

*Ramsey theory*by Ronald Graham, Bruce Rothschild, and Joel Spencer [library]*Enumerative combinatorics, volume I*by Richard Stanley [library]*Concrete mathematics*by Ronald Graham, Donald Knuth, and Oren Patashnik [library]*Generatingfunctionology*by Herbert Wilf [library] [online]*Analytic combinatorics*by Philippe Flajolet, and Robert Sedgewick [library] [online]*Probabilistic method*by Noga Alon, and Joel Spencer [library]*Probability Theory and Combinatorial Optimization*by Michael Steele for Azuma trickery [video]*Graph theory*by Reinhard Diestel [library] [online]*Sumsets and structure*(chapter) by Imre Ruzsa in*Combinatorial number theory and additive group theory*[library]*Additive combinatorics*by Terence Tao, and Van Vu [library]

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

#### More fun:

More fun can be had at my *office hours* on Mondays 4:30–5:30pm and Thursdays 10:30–11:30am 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.

#### Lectures:

- August 26: Introduction. Dirichlet's approximation theorem.
- August 28: Erdős–Szekeres on monotone sequences. Ramsey's theorem. Schur's theorem. Upper and lower bounds on the Ramsey function.
- August 30: About bounds on the Ramsey function, and probabilistic arguments. Hypergraphs. Ramsey's theorem for hypergraphs. Erdős–Szekeres on points in convex position.
- September 2:
*Labor day* - September 4: Ramsey's theorem for hypergraphs (edge version). Statement of stepping-up lemma. Homework #1
- September 6: Stepping-up lemma. Compactness principle (statement).
- September 9: Compactness principle. Chromatic number of the plane. Notes on the compactness principle.
- September 11: Infinite Ramsey theorem. van der Waerden's theorem (part I).
- September 13: van der Waerden's theorem (part II).
- September 16: van der Waerden's theorem (part III). Statement of Hales–Jewett theorem. Tic-tac-toe.
- September 18: Hales–Jewett theorem. Gallai's theorem. Cubes in sets of positive density.
- September 20: Szemerédi's proof of Roth's theorem (part I).
- September 23: Szemerédi's proof of Roth's theorem (part II). Homework #2
- September 25: Binomial coefficients. Stirling numbers of the second kind. Binomial theorem. Changemaking in Fictionland.
- September 27: Monetary reform in Fictionland. Generating functions. Vandermonde convolution.
- September 30: Alternating permutations (part I). Complex-analytic marvels.
- October 2: Alternating permutations (part II).
- October 4: Catalan numbers.
- October 7: Dyck paths. Bijection for Catalan numbers. Fibonacci without rabbits. Pringsheim's theorem (statement).
- October 9: Pringsheim's theorem (proof). Balanced trees. Homework #3
- October 11: Balanced trees. Domino tilings at large (part I).
- October 14: Domino tilings at large (part II).
- October 16: Linearity of expectation. Bipartite subgraphs. LYM inequality.
- October 18:
*Mid-semester break* - October 21: Sum-free subsets. Alterations. Dominating sets. Independent sets. Homework #4
- October 23: Property B (part I).
- October 25: Property B (part II).
- October 28: Conditional expectation. Variance and Chebyshev's inequality. Concentration of binomial distribution. Subgraph appearance threshold (part I).
- October 30: Subgraph appearance threshold (part II). Distinct sums problem. Motivation for the local lemma.
- November 1: Local lemma. Property B in sparse hypergraphs.
- November 4: Decomposing packings. Latin transversals. Homework #5
- November 6: Latin transversals (cont'd). Chernoff's inequality.
- November 8: Martingales. Azuma's inequality.
- November 11: General exposure martingales. Concentration of the chromatic number.
- November 13: Mean of the chromatic number (part I). Notes on the chromatic number of a random graph
- November 15: Mean of the chromatic number (part II).
*Guest lecture by Po-Shen Loh* - November 18: Approximate isoperimetric inequality on the hypercube. Szemerédi's regularity lemma (motivation). Notes on the regularity lemma
- November 20: Szemerédi's regularity lemma (statement + increment step). Homework #6
- November 23: Szemerédi's regularity lemma (end of the proof).
- November 25: Triangle removal lemma. Another proof of Roth's theorem.
- November 27:
*Thanksgiving (part I)* - November 29:
*Thanksgiving (part II)* - December 2: Behrend's construction.
- December 4: Sum-product theorem (part I).
- December 6: Sum-product theorem (part II). Triangle inequalities.