﻿ Algebraic Combinatorics I

# Cezar Lupu

## Course Description

Combinatorics is a fascinatingly large field of mathematics. Typically, it deals with finite structures, but it is also characterized by its methods. In this course we will revisit some of the more "classical" topics in combinatorics such as binomials, principle of inclusion-exclusion or graph theory, but we will also look at some new topics such as extremal graph theory/Ramsey theory, intersecting sets and partially ordered sets, the probabilistic method or algebra methods in combinatorics. Moreover, we will dwell on applications of combinatorics in algebra, geometry, analysis or topology.

## Course outline

Chapter 0: What is combinatorics?
Chapter 1: Introduction to combinatorics
Chapter 2: The principle of inclusion-exclusion
Chapter 3: Generating functions
Chapter 4: Intersecting sets and partially ordered sets
Chapter 5: Introduction to graph theory
Chapter 6: Extremal graph theory / Ramsey theory
Chapter 7: The probabilistic method
Chapter 8: Algebraic methods in combinatorics

## Homework (Assigned Problems)

Homework 1-Problems: Example 1.3.11 (Erdos), 1.5.6, 1.5.7, 1.5.8, 1.5.9, 1.5.10, 1.5.11, 1.5.14, 1.5.15, 1.5.18.
Homework 2-Problems: 2.5.2, 2.5.3, 2.5.6, 2.5.12, 2.5.13, 2.5.15
Homework 3-Problems: 3.6.1, 3.6.2, 3.6.3, 3.6.5, 3.6.11, 3.6.14
Homework 4-Problems: 4.9.1, 4.9.2, 4.9.5, 4.9.9, 4.10.20
Homework 5-Problems: 5.6.1, 5.6.2, 5.6.3, 5.6.6
Homework 6-Problems: 6.7.1, 6.7.2, 6.7.3, 6.7.4, 6.7.7

#### Comments and remarks.

1. Make sure you submit your homework as a single PDF file as in the format NAMESTUDENT_AC1_HW#.PDF (where # can be 1,2 ,3, 4, 5, or 6) to the email addresses of my two coaches Tingyu Zhang (email: tyzhang21@mails.tsinghua.edu.cn) and Yigang Zheng (email: zhengyg21@mails.tsinghua.edu.cn)
2. Late homework will NOT be accepted, unless exceptional circumstances.

## Lecture notes and videos.

1. Lecture notes are provided by the instructor via email to the students after each lecture.
2. Video lectures will be provided via email by the instructor of this course.

## Research projects.

Algebraic Combinatorics I List of projects (final list)

1. A list of 10 projects is assigned by the end of the first week of the Mathcamp by the instructor of the course.
2. Research projects are NOT mandatory. Students should decide on their own if he/she wants to complete and create a presentation for the Mathcamp symposium which will happen in the last 2-3 days of the camp.
3. Students should declare their choice on the project by the end of the 2nd week (July 22nd, 12:00 PM) of the Mathcamp.