### Walk through Combinatorics, Fall 2017

#### When:

Mondays, Wednesdays, Fridays 9:30

#### Where:

Baker Hall 235B [map]

#### 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 three 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.

#### 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]

#### More fun:

More fun can be had at my office hours on Wednesdays 2:30–3:30pm and Thursdays 10:30–11:30am in Wean 6202. I am also available by appointment.

#### Course activities:

Mastery of combinatorics requires practice. Hence, there will be regular homeworks. For your own good, you are strongly encouraged to do as much homework as possible individually. Collaboration and use of external sources are permitted, but must be fully acknowledged and cited. All the writing must be done individually. Failure to do so will be treated as cheating. 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 homework will count for 50% of the grade. During the semester there will be two tests (on October 25, and December 6). Each test will count for 25% of the grade. There will be no final exam.

Homework must be submitted in LaTeX via e-mail. I want both the LaTeX file and the PDF that is produced from it. The filenames must be of the form lastname_discr_homeworknumber.tex and lastname_discr_homeworknumber.pdf respectively. Pictures do not have to be typeset; a legible photograph of a hand-drawn picture is acceptable.

The homework must be submitted by 9:30am of the day it is due. For each minute that it is late, the grade will be reduced by 10%.

There will be opportunities for extra credit. Unusually insightful solutions, and other achievements will be appropriately rewarded.

#### Staying sane and healthy:

This is a graduate course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights. Start the assignments early, and get good nutrition and exercise. Pace yourself, for semester is long. If you find yourself falling behind or constantly tired, talk to me.

#### Lectures:

• August 28: Introduction. Pigeonhole principle. Dirichlet's approximation theorem.
• August 30: Erdős–Szekeres on monotone sequences. Ramsey's theorem. Bounds on the middle binomial coefficient. Stirling's formula (statement). Homework #1
• September 1: Schur's theorem. Lower bounds on Ramsey numbers. About bounds on the Ramsey function, and probabilistic arguments. Two-source extractors and Ramsey graphs (diversion). Hypergraphs.
• September 4: Labor Day
• September 6: Ramsey's theorem for hypergraphs (vertex and edge versions). Homework #2
• September 8: Bounds on hypergraph Ramsey numbers. Radon's lemma in the plane. Erdős–Szekeres on points in convex position.
• September 11: Stepping-up lemma. Compactness principle (statement). Chromatic number of the plane.
• September 13: Compactness principle. Notes on the compactness principle.
• September 15: Infinite Ramsey theorem. van der Waerden's theorem (part I).
• September 18: van der Waerden's theorem (part II).
• September 20: van der Waerden's theorem (part III). Statement of Hales–Jewett theorem. Tic-tac-toe. Homework #3
• September 22: Proof sketch of Hales–Jewett theorem. Gallai's theorem. Szemerédi's proof of Roth's theorem (part I).
• September 25: Szemerédi's proof of Roth's theorem (part II).
• September 27: Szemerédi's proof of Roth's theorem (part III).
• September 29: Behrend's construction. Binomial coefficients. Binomial theorem. Changemaking in Fictionland.
• October 2: Monetary reform in Fictionland. Partial fraction decomposition. Generating functions and formal power series.
• October 4: Weak compositions. Solving linear recurrences. Alternating permutations. Homework #4
• October 6: Review of complex analysis (part I): differentiability and contour integration.
• October 9: Review of complex analysis (part II): Cauchy's theorem.
• October 11: Review of complex analysis (part III): Taylor's series. Asymptotics for the number of alternating permutations.
• October 13: Better asymptotics for the number of alternating permutations. Catalan numbers. Dyck paths. Homework #5
• October 16: Bijection for Catalan numbers. Fibonacci without rabbits.
• October 18: Pringsheim's theorem. Balanced trees (part I).
• October 20: Mid-semester break
• October 23: Balanced trees (part II). Domino tilings at large (part I).
• October 25: Test
• October 27: Domino tilings at large (part II). Homework #6
• October 29: Domino tilings at large (part III). Probability spaces. Random variables. Linearity of expectation. Bipartite subgraphs.
• November 1: LYM inequality. Sum-free subsets (several approaches).
• November 3: Dominating sets. Independent sets. Graphs of large girth and large chromatic number (part I).
• November 6: Graphs of large girth and large chromatic number (part II). Property B (part I). Homework #7
• November 8: Property B (part II).
• November 10: No class
• November 13: Second moment. Subgraph appearance threshold (part I). Guest lecture by Wesley Pegden
• November 15: Subgraph appearance threshold (part II). Guest lecture by Wesley Pegden
• November 17: Chernoff's inequality. Existence of codes. Guest lecture by Wesley Pegden
• November 20: Conditional expectation. Martingales. Edge exposure martingales. Azuma's inequality. Homework #8
• November 27: Exposure martingales. Concentration of chromatic number. Notes on exposure martingales.
• November 29: More applications of Azuma's inequality. Isoperimetric problem. Motivation for the local lemma. (Optional: Notes on the chromatic number of a random graph)
• December 1: Local lemma. Property B.
• December 4: Szemerédi's regularity lemma (part I).
• December 6: Test
• December 8: Szemerédi's regularity lemma (part II).

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