Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites
((18.701 Algebra I or 18.703 Modern Algebra) and (18.100A Real Analysis, 18.100B Real Analysis, 18.100P Real Analysis, or 18.100Q Real Analysis)) or permission of instructor
Course Description
This course examines classical and modern developments in graph theory and additive combinatorics, with a focus on topics and themes that connect the two subjects. The course also introduces students to current research topics and open problems.
A foundational result in additive combinatorics is Roth’s theorem, which says that every subset of {1, 2, …, n} without a 3-term arithmetic progression contains o(N) elements. You will see a couple of different proofs of Roth’s theorem: (1) a graph theoretic approach and (2) Roth’s original Fourier analytic approach. A central idea in both approaches is the dichotomy of structure versus pseudorandomness, and it is one of the key themes of the course.
Roth’s theorem laid the groundwork for many important later developments, such as the following:
- Szemerédi’s theorem. Every set of integers of positive density contains arbitrarily long arithmetic progressions.
- Green–Tao theorem. The primes contain arbitrarily long arithmetic progressions.
The course will explore these and related topics, including the following:
- Extremal graph theory. What is the maximum number of edges in a triangle-free graph on n vertices? What if instead we forbid cycles of length 4? At most how many edges can an n-vertex graph have if every edge is contained in exactly one triangle?
- Szemerédi’s regularity lemma. A powerful tool in combinatorics that provides an approximate description/decomposition for every large graph.
- Pseudorandom graphs. What does it mean for some graph to resemble a random graph?
- Graph limits. In what sense can a sequence of graphs, increasing in size, converge to some limit object?
- Fourier analysis in additive combinatorics. An important technique for studying combinatorial properties in arithmetic settings, e.g., in the proof of Roth’s theorem.
- Freiman’s theorem and the structure of sum sets. What can one say about sets \(A\) such that the sum set \(A+A= {a+a’ : a,a’ \in A }\) is small?
- Sum-product phenomenon. Can a set \(A\) simultaneously have both small sum set \(A+A\) and product set \(A \cdot A\)?
Although the course will be largely divided into two parts (graph theory in the first half and additive combinatorics in the second), we will emphasize the interactions between the two topics and highlight the common themes.
Textbook
Zhao, Yufei. Graph Theory and Additive Combinatorics: Exploring Structure and Randomness. Cambridge University Press, 2023. ISBN: 9781009310949.
Grading
- Primarily based on problem sets
- Only non-starred problems are considered for the calculations of letter grades other than A and A+.
- The final grade may be adjusted up to ±5% at the end of term holistically based on the following:
- Research project proposal assignment
- Participation (in class, office hours, and on Piazza)
- Final letter grade cutoffs:
- 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+.
- Official MIT grading policy: ± grade modifiers do not count towards the GPA and do not appear on the external transcript.
