图论与组合 (Fall 2011)

Table of Contents

News

  • 2011/12/22: Office Hour 周一晚7到9点在俞老师办公室，需要确认期末考成绩的同学可以来看考卷。我感觉在grading的时候都看得很清楚了，成绩不会有什么变动。
  • 2011/12/22: 请下列同学务必来Office Hour（考卷签名问题）：张舒畅、肖长城、王朝阳、肖涛、孔维昊、寿鹤鸣。
  • 2011/12/22: 期末考试参考答案和简要小结在下面的Exam section。为了确认你的成绩和预期的相符，你可以：(a)写邮件告诉我期末的估算成绩并询问成绩。(b)周一晚上来office hour看考卷。如果周一不方便的话可以email单独约。
  • 2011/12/17: Hw12，文件中加了参考答案。
  • 2011/12/14: Hw11，更新了答案。
  • 2011/11/15: 公布了期中考试的参考答案。
  • 2011/10/20: 添加了和近期课程有关的推荐读物。

Lecture Notes

下面是简要的讲义。希望大家积极提供反馈，补充和指正。

(ps) - (pdf)

  1. Lecture 1, 2011/09/07 Basic counting, binomial numbers.
  2. Lecture 2, 2011/09/14 Combinatorial proofs, Catalan numbers.
  3. Lecture 3, 2011/09/21 Principle of Inclusion-Exclusion, Mobius inversion.
  4. Lecture 4, 2011/09/26 Mobius inversion on numbers, and a little preview of incidence algebra.
  5. Lecture 5, 2011/09/28 Stirling numbers.
  6. Lecture 6, 2011/10/10 Basic concepts in graphs.
  7. Lecture 7, 2011/10/12 More basic concepts in graphs.
  8. Lecture 8, 2011/10/19 The pigeon hole principle, the Ramsey principle.
  9. Lecture 9, 2011/10/24 Problems from the homeworks; Hypergraphs, Ramsey theorem, Erdos-Szekeres theorem, and Erdos' theorem on sum-free subsets.
  10. Lecture 10, 2011/11/02 Problems from the homeworks and the midterm; The probabilistic method, linearity of expectation.
  11. Lecture 11, 2011/11/07 Probability methods, alterations. Ramsey numbers. Graphs with big girth and big chromatic number.
  12. Lecture 12, 2011/11/09 Probability methods, property B.
  13. Lecture 13, 2011/11/16 Independent events. The Lovasz local lemma.
  14. Lecture 14, 2011/11/21 Using the Lovasz local lemma. Mantel's theorem, Turan graphs.
  15. Lecture 15, 2011/11/23 Extremal graph theory: Turan's theorem, Hamilton cycles, girth >=5 graphs, Erdos-Stone theorem.
  16. Lecture 16, 2011/11/30 Extremal graph theory: Szemeredi regularity lemma and its applications.
  17. Lecture 17, 2011/12/05 Hall's marriage theorem, Dilworth theorem.
  18. Lecture 18, 2011/12/07 Sperner's theorem, Erdos-Ko-Rado theorem. Basic linear algebraic methods.
  19. Lecture 19, 2011/12/14 More linear algebraic topics. Spectrum of graphs. Lindstrom lemma, Cauchy-Binet formula, and Matrix tree theorem.
  20. Lecture 20, 2011/12/19 Problems in combinatorial geometry.

Homework

  1. Homework1 : (tex) - (ps) - (pdf)
  2. Homework2 : (tex) - (ps) - (pdf)
  3. Homework3 : (tex) - (ps) - (pdf)
  4. Homework4 : (tex) - (ps) - (pdf)
  5. Homework5 : (tex) - (ps) - (pdf)
  6. Homework6 : (tex) - (ps) - (pdf)
  7. Homework7 : (tex) - (ps) - (pdf)
  8. Homework8 : (tex) - (ps) - (pdf)
  9. Homework9 : (tex) - (ps) - (pdf)
  10. Homework10 : (tex) - (ps) - (pdf)
  11. Homework11 with solutions : (tex) - (ps) - (pdf)
  12. Homework12 with solutions : (tex) - (ps) - (pdf)

Exams

Midterm with solutions to section II: (ps) - (pdf)

Final with solutions and remarks: (ps) - (pdf) (I will update this with students' nice solutions and some statistics by Monday.)

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 Erdos and his co-authors, and attack open problems. The course will be self-contained. The students are assumed to have the basic ability in problem solving.

Textbook(s) and Articles

教科书：

  • J.H. van Lint and R. M. Wilson: A Course in Combinatorics (2nd ed). Combridge University Press, 2001.

推荐读物：

  • B. Bollobas: Combinatorics. Combridge University Press, 1986.
  • R. Stanley: Enumerative Combinatorics Vol 1, Combridge 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.
  • Articles about Paul Erdos: http://www.ams.org/notices/199801/comm-erdos.pdf

Contact

陈晓敏 gougle [at] gmail [dot] com

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