# 18.226 Probabilistic Methods in Combinatorics

**Fall 2024, MIT**

**Class meetings:** Mondays and Wednesdays 2:30–4pm, room 45-102

*(First class Wednesday September 4)*

**Instructor:** Sammy Luo

**Graders:** Hang Du and Honglin Zhu

**Office Hours:** After class (MW 4-4:45pm) in 45-102; Friday 2:30-3:30pm in 2-132

- You may ask me about unstarred problems (but not starred ones) during office hours.
- I will ask you to first explain on the board the progress you have made so far.

**Email policy**

- My policy is to not answer by email any math questions related to the class, due to time constraints and also the fact that emails are not a great medium for such Q&As (ask them during office hours instead).
- Email the graders and cc me for anything homework-related (submission, extensions, grading, etc.)
- Begin your email subject line with “[18.226]”

## Course description and policies

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

Topics:

- Introduction
- Linearity of expectations
- Alterations
- Second moment
- Chernoff bound
- Lovász local lemma
- Correlation inequalities
- Janson inequalities
- Concentration of measure
- Entropy
- Container

*Textbook:* Alon and Spencer, The probabilistic method, Wiley (the latest edition is 4th, but earlier editions suffice). Available electronically from MIT Libraries.

We will be closely following Prof. Yufei Zhao's lecture notes.

*Prerequisites:* Mathematical maturity at the level of a first-year math graduate student.
Comfort with combinatorics (18.211), probability (18.600), and real analysis (18.100).

*Grading:* 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, at my discretion 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 my discretion based on overall performance. Solving a significant number of starred problems is a requirement for grades of A and A+ (roughly 1/3 of the problems for an A, and 2/3 for an A+, up to adjustments for the difficulty of individual problems).

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

If you wish to attend lectures, please register as for credit or listener.

Students needing support should consider reaching out to Student Support Services (S^{3}), GradSupport, or Student Disability Services.

### Lectures

- 9/4 Introduction
- 9/9 Introduction
- 9/11 Linearity of expectations
- 9/16 Linearity of expectations
- 9/18 Alterations
- 9/23 Second moment
- 9/25 Thresholds
- 9/30 Second moment
- 10/2 Chernoff bound
- 10/7 Lovász local lemma
- 10/9 Lovász local lemma
*10/14 Holiday (Indigenous Peoples' Day). No class*- 10/16 Lovász local lemma
- 10/21 Lovász local lemma
- 10/23 Correlation inequalities
- 10/28 Janson inequalities
- 10/30 Janson inequalities
- 11/4 Concentration of measure
- 11/6 Concentration of measure
*11/11 Holiday (Veterans Day). No class*- 11/13 Concentration of measure
- 11/18 Concentration of measure
- 11/20 Concentration of measure
- 11/25 Concentration of measure
*11/27 Class canceled*- 12/2 Entropy method
- 12/4 Entropy method
- 12/9 Entropy method
- 12/11 Container method

## Homework

The problem set will be updated over the course of the semester. I will announce on Canvas when each problem set is complete.

You should only submit the designated problems, but are encouraged to try the rest as well.

Starred problems are generally more challenging.

To get the most out of this course, you are expected to spend a significant amount of time solving these problems. It will be essential to start thinking about these problems early.

Problem set | Due date |
---|---|

PS 1 | Sun Sep 22 |

PS 2 | Sun Oct 6 |

PS 3 | Sun Oct 20 |

PS 4 | Sun Nov 3 |

PS 5 | Sun Nov 17 |

PS 6 | Sun Dec 8 |

**Starred problems** on each problem set should be
submitted separately, with an automatic 7-day extension without penalty
(and no additional lateness permitted).

### Submissions

- Must be typed in LaTeX and submitted as PDF on Gradescope (accessed from Canvas).
- Due time:
**11:59pm of each due date** - Begin each solution on a new page
- Select the pages for each problem on Gradescope (or else they will not be graded)
- Each box on the left-margin of the problem set PDF indicates a single problem worth 10 points
- Regrade requests should be submitted on Gradescope within one week of the grades being published

### Late policy

**Penalty.**Late submissions will be penalized by 20% per each late day (24-hour increments, without fractional accounting).- Example: for an assignment due on Sunday, a submission worth
*x*points if turned in on time will be worth 0.6*x*points if submitted on Tuesday.

- Example: for an assignment due on Sunday, a submission worth
**Multiple submissions.**You are allowed to submit some of the problems on time and some of the problems in a separate “late” batch and have the late penalty applied only to the late batch.- If you take this option, you must (1) email the graders explaining which problems were submitted on time and which ones were late and by how many dates, and (2) clearly indicate this information also in your write-up.
- Multiple submissions complicate our grading process. If you do not do the above, your entire submission may receive a late penalty as if all problems were submitted by the last submission time.
- You are allowed to submit at most two batches (i.e., 1 ontime + 1 late batch, or 2 separate late batches)
- The second batch may not contain any updates or replacement to an already submitted problem.
- The final submission should include everything, with each problem’s lateness clearly marked in the PDF.
- This policy is provided as a courtesy. It may be rescinded if abused.

**Extensions.**If you need an extension for valid excuses (e.g., unanticipated health or family issues), please email the graders and me**in advance**or have S3 send us a message. Let us know how many days extension you need.- My policy is to not grant extension based on foreseeable circumstances including academic workload, extracurriculars, and poor study habits.

- If you submit your homework late, you should not discuss the homework with other students who have already submitted their homework until you have submitted yourself.

### Collaborations

- You are encouraged to first work on the homework problems independently before seeking collaboration.
- Meaningful collaboration is allowed if it helps with your learning (e.g., solving a problem together)
- Unacceptable practices include: “dividing up” the problems among a group and then distributing the solutions; asking for a solution from a friend.
- You must write up your own solutions.
- Pset partners — a tool for finding problem set collaborators (MIT Touchstone required)

### Acknowledging collaborators and sources

It is required to acknowledge your sources (even if you worked independently)

- At the beginning of the submission for
**each problem**, write`Collaborators and sources:`

followed by a list of collaborators and sources consulted (people, books, papers, websites, software, etc.), or write`none`

if you did not use any such resources. - Failure to acknowledge will result in an automatic 1pt penalty per problem.
- Acceptable uses of resources include: looking up a standard theorem/formula/technique; using Wolfram Alpha/Mathematica/Python for a calculation (no need to mention lecture notes or textbook)
- Unacceptable uses of resources include: directly looking up the problem online or in the research literature for a solution. (Once you have solved a problem, it is fine to seek and learn alternate solutions.)

Intentional violations of the above policies may be considered academic dishonesty/misconduct.

### Additional resources

- Richard P. Stanley Seminar in Combinatorics – research seminar covering recent research results by various speakers. Intended for graduate students and researchers in combinatorics, but all are welcome. Self-sign up for mailing list.

**Previous version of the class taught by Prof. Yufei Zhao:**
Spring 2019,
Fall 2020,
Fall 2022.