18.226 | Fall 2022 | Graduate

Probabilistic Methods in Combinatorics

Pages

Lecture 1: Introduction

Lecture 2: Introduction (cont.)

Lecture 3: Linearity of Expectations

Lecture 4: Linearity of Expectations (cont.)

Lecture 5: Alterations

Problem set 1 due

Lecture 6: Second Moment

Lecture 7: Second Moment (cont.)

Lecture 8: Chernoff Bound

Lecture 9: Chernoff Bound (cont.)

Problem set 2 due

Lecture 10: Lovász Local Lemma

Lecture 11: Lovász Local Lemma (cont.)

Lecture 12: Special lecture by Joel Spencer

Problem set 3 due

Lecture 13: Lovász Local Lemma (cont.)

Lecture 14: Lovász Local Lemma (cont.)

Lecture 15: Correlation Inequalities

Lecture 16: Janson Inequalities

Problem set 4 due

Lecture 17: Janson Inequalities (cont.)

Lecture 18: Concentration of Measure

Lecture 19: Concentration of Measure (cont.)

Lecture 20: Concentration of Measure (cont.)

Problem set 5 due

Lecture 21: Concentration of Measure (cont.)

Lecture 22: Concentration of Measure (cont.)

Lecture 23: Concentration of Measure (cont.)

Lecture 24: Entropy 

Lecture 25: Entropy (cont.)

Problem set 6 due

Lecture 26: Entropy (cont.)

Lecture 27: Containers

Instructor Interview

Below, Professor Yufei Zhao describes various aspects of how he taught 18.226 Probabilistic Methods in Combinatorics.

OCW: You structured problem sets a little differently in this course, providing students with a single file with many problems but only requiring a subset of these problems to be turned in for assessment. Tell us about your decision to structure problem sets this way.

Yufei Zhao: I wanted to give the students lots of opportunities to practice the techniques taught in lectures. The single-file format made it easier to add new problems to the list as the semester progressed, and in a way that was synchronized with the pace of the lectures. I only required the students to turn in a subset of the problems in order to ease the workload burden.

OCW: How did you use the lecture time in class to ensure maximum benefit to your students?

Yufei Zhao: The goal of the course was to introduce students to the probabilistic method, which has many applications that are quite nice and short. The students appreciated seeing a lot of examples in class. Longer proofs tend to be less well suited to the lecture format, as they’re harder to follow. I’ve found that the lectures that work best are ones filled with small and neat examples and applications illustrating the method being discussed.

Curriculum Information

Prerequisites

or permission of instructor

Requirements Satisfied

18.226 can be applied toward a doctorate degree in Pure or Applied Mathematics, but is not required.

Offered

Every other fall semester. 

Assessment

Grade Breakdown

Grading for 18.226 was primarily based on homework grades (no exams). A modifier up to 5 percentage points may be applied (in either direction) in calculating the final grade, based on factors such as participation.

Student Information

Enrollment

41 students

Breakdown by Year

The class included a mixture of graduate students and undergraduates.

Breakdown by Major

Students in the course came primarily from mathematics and computer science.

Typical Student Background

Most students in the course had strong mathematical problem solving backgrounds; many of the undergraduates had experience participating in math competitions.

How Student Time Was Spent

During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:

Lecture

Met 2 times per week for 1.5 hours per session; 25 sessions total.

Out of Class

Outside of class, students spent most of their time solving problems from the six assigned problem sets.

The problem sets (PDF) are a list of problems for practice. A subset of these problems will be designated as to-be-turned-in. Only the designated problems are required to be submitted. Bonus problems, marked by ★, are more challenging. A grade of A- may be attained by only solving the non-starred problems. To attain a grade of A or A+, you should solve a substantial number of starred problems.

PROBLEM SET PROBLEMS
Problem set 1 A3, A8, A9, A10★, A11 (a, b★), A12★, A14, A15, A16★, and A17★
Problem set 2 B4, B5, B7★, B8★, B9★, C1, C3, C4, C6★, and C7
Problem set 3 C8, C10, C11★, C12★, D2–D4, D5 (a★, b★), E1, and E3★
Problem set 4 E4, E6, E7, E8★, E9★, E10(a, b★), E11(d★), E12, and E13★
Problem set 5 F3 (a, b), F6, F7★, F8★, G1, G3 (a), G4★, G6★, H1, H4★, and H5★
Problem set 6 H7★, H9★, H10, H11, H13, H14★, I3★, I4, I5, I6★, I7★, and I10★

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Prerequisites

or permission of instructor

Course Description

This course is a graduate-level introduction to the probabilistic methods, a fundamental and powerful technique in combinatorics and theoretical computer science. The essence of the approach is to show that some combinatorial object exists and prove that a certain random construction works with positive probability. The course focuses on methodology as well as combinatorial applications.

Topics

  • Linearity of expectations
  • Alterations
  • Second moment
  • Chernoff bound
  • Lovász local lemma
  • Correlation inequalities
  • Janson inequalities
  • Concentration of measure
  • Entropy
  • Containers

Textbook

Alon, Noga and Joel H. Spencer. The Probabilistic Method. Wiley, 2016. ISBN: 9781119061953. (The fourth edition is the latest, but earlier editions suffice.)

Grading

Grading is primarily based on homework grades (no exams). A modifier up to 5 percentage points may be applied (in either direction) in calculating the final grade, based on factors such as participation.

Final letter grade cutoffs: Only non-starred problems are considered for the calculations of letter grades other than A and A+.

  • A− : ≥ 85%
  • B− : ≥ 70%
  • C− : ≥ 50%

Grades of A and A+ are awarded at instructor’s discretion based on overall performance. Solving a significant number of starred problems is a requirement for grades of A and A+.

Note that for MIT students, ± grade modifiers do not count towards the GPA and do not appear on the external transcript.

Course Info

Instructor
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As Taught In
Fall 2022
Level
Learning Resource Types
Lecture Notes
Instructor Insights
Problem Sets