GRAPHTHEORY AND ADDITIVECOMBINATORICS notes for mit 18.217 (fall 2019) lecturer: yufei zhao http://yufeizhao.com/gtac/ About this document This document contains the course notes for Graph Theory and Additive Combinatorics, a graduate-level course taught by Prof. Yufei Zhao at MIT in Fall 2019. The notes were written by the students of the class based on the lectures, and edited with the help of the professor. The notes have not been thoroughly checked for accuracy, espe- cially attributions of results. They are intended to serve as study resources and not as a substitute for professionally prepared publica- tions. We apologize for any inadvertent inaccuracies or misrepresen- tations. More information about the course, including problem sets and lecture videos (to appear), can be found on the course website: http://yufeizhao.com/gtac/ Contents A guide to editing this document 7 1 Introduction 13 1.1 Schur’s theorem........................ 13 1.2 Highlights from additive combinatorics.......... 15 1.3 What’s next?.......................... 18 I Graph theory 21 2 Forbidding a subgraph 23 2.1 Mantel’s theorem: forbidding a triangle.......... 23 2.2 Turán’s theorem: forbidding a clique............ 24 2.3 Hypergraph Turán problem................. 26 2.4 Erd˝os–Stone–Simonovits theorem (statement): forbidding a general subgraph...................... 27 2.5 K˝ovári–Sós–Turán theorem: forbidding a complete bipar- tite graph............................ 28 2.6 Lower bounds: randomized constructions......... 31 2.7 Lower bounds: algebraic constructions.......... 34 2.8 Lower bounds: randomized algebraic constructions... 37 2.9 Forbidding a sparse bipartite graph............ 40 3 Szemerédi’s regularity lemma 49 3.1 Statement and proof..................... 49 3.2 Triangle counting and removal lemmas.......... 53 3.3 Roth’s theorem........................ 58 3.4 Constructing sets without 3-term arithmetic progressions 59 3.5 Graph embedding, counting and removal lemmas.... 61 3.6 Induced graph removal lemma............... 65 3.7 Property testing........................ 69 3.8 Hypergraph removal lemma................. 70 3.9 Hypergraph regularity.................... 71 3.10 Spectral proof of Szemerédi regularity lemma...... 74 4 4 Pseudorandom graphs 77 4.1 Quasirandom graphs..................... 77 4.2 Expander mixing lemma................... 82 4.3 Quasirandom Cayley graphs................ 84 4.4 Alon–Boppana bound.................... 86 4.5 Ramanujan graphs...................... 88 4.6 Sparse graph regularity and the Green–Tao theorem.. 89 5 Graph limits 95 5.1 Introduction and statements of main results....... 95 5.2 W-random graphs....................... 99 5.3 Regularity and counting lemmas.............. 100 5.4 Compactness of the space of graphons........... 103 5.5 Applications of compactness................ 106 5.6 Inequalities between subgraph densities.......... 110 II Additive combinatorics 119 6 Roth’s theorem 121 6.1 Roth’s theorem in finite fields................ 121 6.2 Roth’s proof of Roth’s theorem in the integers...... 126 6.3 The polynomial method proof of Roth’s theorem in the fi- nite field model........................ 132 6.4 Roth’s theorem with popular differences......... 137 7 Structure of set addition 141 7.1 Structure of sets with small doubling........... 141 7.2 Plünnecke–Ruzsa inequality................. 144 7.3 Freiman’s theorem over finite fields............ 147 7.4 Freiman homomorphisms.................. 149 7.5 Modeling lemma....................... 150 7.6 Bogolyubov’s lemma..................... 153 7.7 Geometry of numbers.................... 156 7.8 Proof of Freiman’s theorem................. 158 7.9 Freiman’s theorem for general abelian groups...... 160 7.10 The Freiman problem in nonabelian groups........ 161 7.11 Polynomial Freiman–Ruzsa conjecture........... 163 7.12 Additive energy and the Balog–Szémeredi–Gowers theo- rem............................... 165 8 The sum-product problem 171 8.1 Crossing number inequality................. 171 8.2 Incidence geometry...................... 172 8.3 Sum-product via multiplicative energy.......... 174 Sign-up sheet Please sign up here for writing lecture notes. Some lectures can be covered by two students working in collaboration, depending on class enrollment. Please coordinate among yourselves. When editing this page, follow your name by your MIT email formatted using \email{[email protected]}. 1. 9/9: Yufei Zhao [email protected] 2. 9/11: Anlong Chua [email protected] & Chris Xu [email protected] 3. 9/16: Yinzhan Xu [email protected] & Jiyang Gao [email protected] 4. 9/18: Michael Ma [email protected] 5. 9/23: Hung-Hsun Yu [email protected] & Zixuan Xu [email protected] 6. 9/25: Tristan Shin [email protected] 7. 9/30: Shyan Akmal [email protected] 8. 10/2: Lingxian Zhang [email protected] & Shengwen Gan [email protected] 9. 10/7: Kaarel Haenni [email protected] 10. 10/9: Sujay Kazi [email protected] 11. 10/16: Richard Yi [email protected] 12. 10/21: Danielle Wang [email protected] 13. 10/23: Milan Haiman [email protected] & Carl Schildkraut [email protected] 14. 10/28: Yuan Yao [email protected] 15. 10/30: Carina Letong Hong [email protected] 16. 11/4: Dhruv Rohatgi [email protected] 17. 11/6: Olga Medrano [email protected] 18. 11/13: Dain Kim [email protected] & Anqi Li [email protected] 19. 11/18: Eshaan Nichani [email protected] 20. 11/20: Alan Peng [email protected] & Swapnil Garg [email protected] 21. 11/25: Adam Ardeishar [email protected] 22. 11/27: Ahmed Zawad Chowdhury [email protected] 23. 12/2: Allen Liu [email protected] 24. 12/4: Mihir Singhal [email protected] & Keiran Lewellen [email protected] 25. 12/9: Maya Sankar [email protected] 26. 12/11: Daishi Kiyohara [email protected] A guide to editing this document Please read this section carefully. Expectations and timeline Everyone enrolled in the course for credit should sign up to write notes for a lecture (possibly pairing up depending on enrollment) by editing the signup.tex file. Please sign up on Overleaf using your real name (so that we can see who is editing what). You can gain read/write access to these files from the URL I emailed to the class or by accessing the link in Stellar. The URL from the course website does not allow editing. All class participants are expected and encouraged to contribute to editing the notes for typos and general improvements. Responsibilities for writers By the end of the day after the lecture, you should put i.e., by Tuesday night for Monday lec- up a rough draft of the lecture notes that should, at the minimum, tures and Thursday night for Wednes- day lectures include all theorem statements as well as bare bone outlines of proofs discussed in lecture. This will be helpful for the note-takers of the following lecture. Within four days of the lecture, you should complete a pol- i.e., by Friday for Monday lectures and ished version of the lecture notes with a quality of exposition similar Sunday for Wednesday lectures to that of the first chapter, including discussions, figures (wherever helpful), and bibliographic references. Please follow the style guide below for consistency. Please note that the written notes are supposed to more than sim- ply a transcript of what was written on the blackboard. It is impor- tant to include discussions and motivations, and have ample “bridge” paragraphs connecting statements of definitions, theorems, proofs, etc. 8 Also, once you have a complete draft, email me at [email protected] (please cc your coauthor) to set up a 30-min appointment to go over your writing. Let me know when you will be available in the upcom- ing three days. At the appointment, ideally within a week of the lecture, please bring a printed copy containing the pages of your writing, and we will go over the notes together for comments. After our one-on-one meeting, you are expected to edit the notes according to feedback as soon as possible while your memory is still fresh, and complete the revision within three days of our meeting. Please email me again when your revision is complete. If the comments are not satisfactorily addressed, then we may need to set up additional appointments, which is not ideal. LATEX style guide Please follow these styles when editing this document. Use lec1.tex as an example. Always make sure that this document compiles without errors! Files Start a new file lec#.tex for each lecture and add \input{lec#} to the main file. Begin the file with the lecture date and your name(s) using the following command. If the file starts a new chapter or sec- tion, then insert the following line right after the \begin{section} or \begin{chapter} or else the label will appear at the wrong location. \dateauthor{9/9}{Yufei Zhao} 9/9: Yufei Zhao English Please use good English and write complete sentences. Never use informal shorthand “blackboard” notation such as ), 9, and 8 in formal writing (unless you are actually writing about math- ematical logic, which we will not do here). Avoid abbreviations such as iff and s.t.. Avoid beginning a sentence with math or numbers. This is a book Treat this document as a book. Do not refer to “lec- tures.” Do not say “last lecture we . .” Do not repeat theorems carried between consecutive lectures. Instead, label theorems and refer to them using \cref. You may need to coordinate with your classmates who wrote up earlier lectures. As you may have guessed, the goal is to eventually turn this docu- ment into a textbook. I thank you in advance for your contributions. Theorems Use Theorem for major standalone results (even if the result is colloquially known as a “lemma”, such as the “triangle re- 9 moval lemma”), Proposition for standalone results of lesser impor- tance, Corollary for direct consequences of earlier statements, and Lemma for statements whose primary purpose is to serve as a step in a larger proof but otherwise do not have major independent interest.