CS477 Combinatorics (Spring 2019)

星期五 12:55-15:40（双周），上院103

News

2019/06/18: So glad to see the wiki resurrects. Our last homework set posted.
2019/05/18: HW10 posted.
2019/05/06: There is no homework this week, but please review the incidence algebra and Mobius inversions. You may want to compute the Mobius function for the poset (X, |), where X is the set of all positive divisors of 72, or 360.
2019/04/27: HW8 posted. Note that our next class will be on 05/05.
2019/04/15: About HW4 P6, my bad that I forgot to specify my requirement -- this is almost the same as that in HW1, but for this one I wanted you work with formal P.I.E. proof.

Lectures and Notes

Lecture 1, 2019/03/01. Appetizers; asymptotics of functions; sets.
Lecture 2, 2019/03/08. Sets, relations, functions, start the binomial numbers.
Lecture 3, 2019/03/15. Binomial numbers and their shapes. The statement and proof of P.I.E.
Lecture 4, 2019/03/22. Applications of P.I.E. Definition of graphs.
Lecture 5, 2019/03/29. Basics of graph theory.
Lecture 6, 2019/04/12. Graphs and colouring. Basics of permutations.
Lecture 7, 2019/04/19. More on permutations. Stirling number of 1st and 2nd kind.
Lecture 8, 2019/04/26. Matchings in bipartite graphs. Definition of posets.
Lecture 9, 2019/05/05. Posets, Dilworth theorem. Incidence algebra and Mobius inversion.
Lecture 10, 2019/05/10. Mobius inversion on product posets, and applications. Illusions of Ramsey theorems.
Lecture 11, 2019/05/17. Ramsey problem on integers. Erdos' proof of Ramsey lower bound.
Lecture 12, 2019/05/26. The probabilistic method.
Lecture 13, 2019/06/14. Erdos' theorem on large chromatic numbers and large girth. Set families. Some combinatorial geometry -- points and lines, Sylvester-Galai, De Bruijn-Erdos.

Office Hour

After class or by appointment.

Homework

You can use either Chinese or English or some Portuguese or some Italian in your homework and exams. The pure reason I don't use Chinese too much in writing is that I am slower in typing Chinese.

Please submit all homework via email to gougle [at] gmail [dot] com. You are encouraged to write your work in LaTeX and submit the resulting pdf file together with the tex file. Writing in LaTeX might be time consuming in the beginning, but it is nice / useful / fun.

It is also fine to submit any other human readable format of your work, e.g. photos of your writing / sketch / painting.

Due to the restriction of the Wiki, I renamed tex files as txt below.

NOTE: By default, all the problems are due by the next lecture. Problems with a (*) can be handed in anytime during the semester, and any problem with a (+) can be handed in no later than two weeks after the next lecture.

Course Description

Description: This course serves as a broad exploration in the field of combinatorics, with a focus on the topics in or related to the theory of graphs and hyper-graphs. The course starts with the basic enumerative combinatorics, including combinatorial proofs in counting, the inclusion-exclusion principle and Mobius inversion, recursion and generating functions. Then we will discuss many interesting topics and techniques, including Ramsey theorems, extremal graph theory, conbinatorial designs, combinatorial geometry, graph matching, connectivity, planarity, and colouring, random graphs, Szemeredi's regularity lemma, the probabilistic method, and the algebraic method. We will adore the legendary Erdős and his co-authors, and hopefully attack open problems. The course will be self-contained. The students are surely assumed to have the basic ability in problem solving.

Textbook(s) and Articles

推荐读物：

J.H. van Lint and R. M. Wilson: A Course in Combinatorics (2nd ed). Cambridge University Press, 2001.
B. Bollobas: Combinatorics. Cambridge University Press, 1986.
R. Stanley: Enumerative Combinatorics Vol 1, Cambridge University Press, 2000.
H. Wilf: Generatingfunctionology, A K Peters, 2006. (Also available online)
R. Graham, B. Rothschild, and J. Spencer: Ramsey Theory (2nd ed). Wiley-Interscience, 1990.
N. Alon and J. Spencer: The probabilistic Method (3rd ed). Wiley-Interscience, 2008.
A. Bondy and U.S.R. Murty: Graph Theory with Applications. Elsevier Science Ltd/North-Holland, 1976.
A. Bondy and U.S.R. Murty: Graph Theory. Springer 2010.
B. Bollobas: Modern Graph Theory. Springer, 1998.
D. West: Introduction to Graph Theory (2nd ed). Prentice Hall, 2000.
M. Aigner, G. Ziegler, and K. Hofmann: Proofs from the BOOK (4th ed). Springer, 2009.
M. Aigner: A Course in Enumeration. Springer, 2007.
Articles about Paul Erdős: http://www.ams.org/notices/199801/comm-erdos.pdf
An interview with Endre Szemeredi: http://www.math.toronto.edu/zsuzsi/research/Szemeredi.pdf
A short article on Vašek Chvátal.

Contact

陈晓敏 Email: gougle [at] gmail [dot] com / Wechat: 1837187082