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Graph Theory and Additive Combinatorics, a Graduate-Level Course Taught by Prof 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 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 subgraphs 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 1 Introduction 1.1 Schur’s theorem In the 1910’s, Schur attempted to prove Fermat’s Last Theorem by Schur (1916) reducing the equation Xn + Yn = Zn modulo a prime p. However, he was unsuccessful. It turns out that, for every positive integer n, the equation has nontrivial solutions mod p for all sufficiently large primes p, which Schur established by proving the following classic result. Theorem 1.1 (Schur’s theorem). If the positive integers are colored with finitely many colors, then there is always a monochromatic solution to x + y = z (i.e., x, y, z all have the same color). We will prove Schur’s theorem shortly. But first, let us show how to deduce the existence of solutions to Xn + Yn ≡ Zn (mod p) using Schur’s theorem. Schur’s theorem is stated above in its “infinitary” (or qualitative) form. It is equivalent to a “finitary” (or quantitative) formulation below. We write [N] := f1, 2, . , Ng. Theorem 1.2 (Schur’s theorem, finitary version). For every positive integer r, there exists a positive integer N = N(r) such that if the elements of [N] are colored with r colors, then there is a monochromatic solution to x + y = z with x, y, z 2 [N]. With the finitary version, we can also ask quantitative questions such as how big does N(r) have to be as a function of r. For most questions of this type, we do not know the answer, even approxi- mately. Let us show that the two formulations, Theorem 1.1 and Theo- rem 1.2, are equivalent. It is clear that the finitary version of Schur’s theorem implies the infinitary version. To see that the infinitary ver- sion implies the finitary version, fix r, and suppose that for every 14 schur’s theorem N there is some coloring fN : [N] ! [r] that avoids monochro- matic solutions to x + y = z. We can take an infinite subsequence of (fN) such that, for every k 2 N, the value of fN(k) stabilizes as N increases along this subsequence. Then the fN’s, along this subse- quence, converges pointwise to some coloring f : N ! [r] avoiding monochromatic solutions to x + y = z, but this contradicts the infini- tary statement. Let us now deduce Schur’s claim about Xn + Yn ≡ Zn (mod p). Theorem 1.3. Let n be a positive integer. For all sufficiently large primes Schur (1916) p, there are X, Y, Z 2 f1, . , p − 1g such that Xn + Yn ≡ Zn (mod p). Proof of Theorem 1.3 assuming Schur’s theorem (Theorem 1.2). We write (Z/pZ)× for the group of nonzero residues mod p under multi- plication. Let H be the subgroup of n-th powers in (Z/pZ)×. The index of H in (Z/pZ)× is at most n. So the cosets of H partition f1, 2, . , p − 1g into at most n sets. By the finitary statement of Schur’s theorem (Theorem 1.2), for p large enough, there is a solu- tion to x + y = z in Z in one of the cosets of H, say aH for some a 2 (Z/pZ)×. Since H consists of n-th powers, we have x = aXn, y = aYn, and z = aZn for some X, Y, Z 2 (Z/pZ)×. Thus aXn + aYn ≡ aZn (mod p). Hence Xn + Yn ≡ Zn (mod p) as desired. Now let us prove Theorem 1.2 by deducing it from a similar sounding result about coloring the edges of a complete graph. The next result is a special case of Ramsey’s theorem. Theorem 1.4. For every positive integer r, there is some integer N = N(r) Ramsey (1929) such that if the edges of KN, the complete graph on N vertices, are colored frank ramsey (1903–1930) had made with r colors, then there is always a monochromatic triangle. major contributions to mathematical logic, philosophy, and economics, 26 Proof. r N( ) = r = before his untimely death at age We use induction on . Clearly 1 3 works for 1. Let after suffering from chronic liver 0 r ≥ 2 and suppose that the claim holds for r − 1 colors with N = N . problems. We will prove that taking N = r(N0 − 1) + 2 works for r colors.. Suppose we color the edges of a complete graph on r(N0 − 1) + 2 vertices using r colors. Pick an arbitrary vertex v. Of the r(N0 − 1) + 1 edges incident to v, by the pigeonhole principle, at least N0 edges in- cident to v have the same color, say red. Let V0 be the vertices joined to v by a red edge. If there is a red edge inside V0, we obtain a red introduction 15 triangle. Otherwise, there are at most r − 1 colors appearing among 0 jV0j ≥ N vertices, and we have a monochromatic triangle by induc- tion. We are now ready to prove Schur’s theorem by setting up a graph whose triangles correspond to solutions to x + y = z, thereby allow- f(k − i) ing us to “transfer” the above result to the integers. [ ] ! [ ] Proof of Schur’s theorem (Theorem 1.2). Let f : N r be a coloring. i f(j − i) j f(k − j) k Color the edges of a complete graph with vertices f1, . , N + 1g by giving the edge fi, jg with i < j the color f(j − i). By Theorem 1.4, if N is large enough, then there is a monochromatic triangle, say on vertices i < j < k. So f(j − i) = f(k − j) = f(k − i). Take x = j − i, y = k − j, and z = k − i. Then f(x) = f(y) = f(z) and x + y = z, as desired. Notice how we solved a number theory problem by moving over to a graph theoretic setup. The Ramsey theorem argument in Theo- rem 1.4 is difficult to do directly inside the integers. Thus we gained greater flexibility by considering graphs. Later on we will see other more sophisticated examples of this idea, where taking a number theoretic problem to the land of graph theory gives us a new perspec- tive.
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