21-228: Discrete Mathematics (Spring 2021)

Po-Shen Loh

Last updated 17 May 2021.


Course Description

Combinatorics, which concerns the study of discrete structures such as sets and networks, is as ancient as humanity's ability to count. Yet although in the beginning, combinatorial problems were solved by pure ingenuity, today that alone is not enough. A rich variety of powerful methods have been developed, often drawing inspiration from other fields such as probability, analysis, algorithms, and even algebra and topology.

This course provides a general introduction to this beautiful subject. Students will learn both classical methods for clever counting, as well as how to apply more modern methods to analyze recursions and sequences. Students will also learn fundamental techniques in graph theory. Throughout the course, students will encounter novel challenges that blur the lines between combinatorics and other subjects, such as number theory, probability, and geometry, so that they develop the skills to creatively combine seeming disparate areas of mathematics.

This course is structured around challenge. Lecture topics are hand-picked to reflect the rather advanced ability level of the general CMU student, and consequently, much of the curriculum sequence is original. Homework and exam problems are particularly difficult, and require creative problem-solving rather than application of learned techniques. To encourage students to truly develop these skills, collaboration is encouraged on homework, and exams (which are non-collaborative) will be open-notes.

The only pre-requisite for the course is one of 21-127 Concepts of Mathematics or 21-128 Mathematical Concepts and Proofs.

Learning Objectives

Students will learn how to:

Learning Resources

The course roughly follows the order of topics in the well-written book Discrete Mathematics, by L. Lovász, J. Pelikán, and K. Vesztergombi. This book is not required for the class, because the level of CMU students is somewhat beyond. To adjust for the level of students at CMU, a substantial amount of extra material (not in the textbook) is covered in class. Students are encouraged to reference the book for further clarification, as it provides a clear presentation of an alternate perspective on the content discussed in class.


The course grade will be calculated by applying a curve to the following calculation, in which at least 25% of students will receive an A grade: Historically, the cutoff for an A grade has been around 66% of the weighted total points. All exams will be open notes / open book.


The only way to learn mathematics is to do mathematics. (Paul Halmos)

In order to encourage students to experiment with the concepts taught in class, homework assignments will be given on alternate weeks. They will be due in class on Fridays, at the beginning of lecture. Each assignment will consist of four to five challenging problems, for which the proof or justification of each answer is more important than actual numerical answer.

Since homework is a learning activity, students are welcome to discuss ideas with each other, although collaboration in the writing stage is not permitted. In other words, please do not look at the actual document that another student is handing in.


There will be 3 in-class exams, each with 5 problems, on Fri Feb 26, Fri Mar 26, and Fri Apr 30. There will also be a comprehensive final exam with 8 problems, at a date and time which will be determined by the registrar. No collaboration is permitted on exams, and students are required to have cameras on during the exam period. Due to time zone issues, people who cannot attend the USA Eastern Time exam will have the opportunity to take a make-up exam with entirely different problems between 9:00pm-9:50pm Eastern Time on that same day.

Other make-up exams will be given only in the case of a documented medical excuse, a university-sanctioned absence (e.g., participation in a varsity sporting event), or a family emergency. However, this must be requested before the official start time of each exam.

Course Policies

Attendance/Participation. Attendance and participation are not part of the course grade, but are strongly encouraged because the curriculum of this class contains many original components.

Academic Integrity and Collaboration. Collaboration as detailed above is permitted only on homework assignments, and not on exams. In the event of academic dishonesty, a score of zero will be assigned, and a communication will be sent to the academic advisor. This policy is motivated by the goal of maintaining a fair environment for all learners.

Late Work. There is no penalty for late homework, except that all homework must be submitted by the final class day of the semester. Any homework submitted after that date receives zero credit. However, students are strongly encouraged to submit homework on time, because this keeps the class in sync for healthy collaboration on homework, and homework is very helpful for exam preparation.

Re-grade Requests. Re-grade requests must be submitted in writing, within one week of receiving the graded assignment. They will be honored, but please note that it is possible for scores to decrease after a re-grade.

Accommodations for students with disabilities. If you have a disability and require accommodations, please contact Catherine Getchell, Director of Disability Resources, 412-268-6121, getchell@cmu.edu. If you have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate.

Statement on Student Wellness. As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and/or reduce your ability to participate in daily activities. CMU services are available, and treatment does work. You can learn more about confidential mental health services available on campus at: http://www.cmu.edu/counseling/. Support is always available (24/7) from Counseling and Psychological Services: 412-268-2922.

Detailed syllabus

The semester's tentative schedule is detailed below.

Week Topic Alternate reference Work due Fri
Week 1
(Mon Feb 1)   
Basic counting; induction Chapter 1; Fwd-backward induction Homework 1
Week 2
(Mon Feb 8)
pigeonhole principle
Chapter 2;
Dirichlet's approximation theorem
(on AoPS)
Week 3
(Mon Feb 15)
Homework 2
Week 4
(Mon Feb 22)
Binomial coefficients Chapter 3; Lucas' Theorem Exam 1 on Fri
Week 5
(Mon Mar 1)
Linear algebra and recurrences Chapter 4; Wikipedia, Croot's notes Homework 3
Week 6
(Mon Mar 8)
Generating functions Generatingfunctionology (free book),
by H. Wilf. Sections 1.1-1.3, 2.1-2.3;
notes from MIT's OpenCourseWare
Week 7
(Mon Mar 15)
More generating functions
(No class Fri Mar 19)
Homework 4
Week 8
(Mon Mar 22)
Introduction to graphs Chapter 7 Exam 2 on Fri
Week 9
(Mon Mar 29)
Trees Chapter 8 Homework 5
Week 10
(Mon Apr 5)
Traveling salesman problem
(No class Mon Apr 5)
Chapter 9
Week 11
(Mon Apr 12)
(No class Fri Apr 16)
Chapter 10 Homework 6
Week 12
(Mon Apr 19)
Combinatorial geometry Chapter 11
Week 13
(Mon Apr 26)
Ramsey theory Alt refs #1 and #2 from B. Sudakov's
Princeton Combinatorics course.
Exam 3 on Fri
Week 14
(Mon May 3)
Planar graphs Chapter 12 Homework 7
Final exam
(Mon May 17)
Main 1pm-4pm: Zoom. Alternate 10pm-1am: Zoom.

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