Some topics in Extremal Combinatorics Hong Liu August 9, 2019 The purpose of this note is to give a touch on some topics in extremal combinatorics. These topics are chosen semi-randomly :) When possible, I try to present the proofs \back- wards", or with some intuitions here and there. The proofs would be a bit longer than usual, but hopefully they look more natural this way. The material covered are based on various notes/books/papers, see main texts for refer- ences. Comments are welcome, if you spot mistakes, please let me know :) 2 Contents 1 Extremal graph theory 5 1.1 Tur´antheorem . .5 1.2 Zykov's symmetrisation . .6 1.3 Erd}os-Stonetheorem . .7 1.4 Stability method . .8 2 Szemer´edi'sregularity lemma and its applications 11 2.1 Taster session . 11 2.2 Formal setup . 12 2.3 Key lemmas . 15 2.3.1 Reduced graph . 15 2.3.2 Embedding lemma . 16 2.3.3 Counting lemma . 16 2.4 Ruzsa-Szemer´editriangle removal lemma . 17 2.4.1 Cleaning the graph G ........................... 17 2.4.2 Proof of triangle removal lemma . 18 2.5 (6; 3)-theorem and Roth's theorem . 18 2.6 Ramsey-Tur´anproblem for K4 .......................... 20 2.7 Chv´atal-R¨odl-Szemer´edi-Trotter theorem . 21 2.8 Spectral proof of regularity lemma . 22 3 Pseudorandomness 27 3.1 Quasirandom graphs . 27 3.1.1 Equivalent definitions of quasirandomness . 28 3.1.2 (Codegree) ) (Induced Subgraph Count). 29 3.1.3 (4-cycle Count) ) strongly regular via Cauchy-Schwarz . 32 3 4 Chapter 1 Extremal graph theory In this chapter, we will discuss the classical Tur´an'stheorem in extremal graph theory and present some standard techniques such as Zykov's symmetrisation, stability method. 1.1 Tur´antheorem Before we start, let us consider the following puzzle. Suppose we have to choose n irrational numbers x1; : : : ; xn. How can we maximise the number of pairs (xi; xj) such that xi + xj is rational? One of the most classical extremal problems, nowadays so-called Tur´an-type problem, is: Problem 1.1.1 (Tur´an-type). How dense a graph can be without containing another (usually small) graph as a subgraph? More specifically, given a graph H, we say a graph G contains a copy of H, or H is a subgraph of G, or H ⊆ G, if there is an injective map ' : V (H) ! V (G) that preserves adjacencies, i.e. for any uv 2 E(H), we have '(u)'(v) 2 E(G). We call such a map an embedding of H in G. We say G is H-free if it does not contain H as a subgraph. If in addition, the map preserves also non-adjacencies, then H is an induced subgraph of G. The main parameter we study for Problem 1.1.1 is the extremal number of H, ex(n; H) = maxfe(G): jGj = n and G is H-freeg; is the maximum size of an n-vertex H-free graph. We call an n-vertex graph G an extremal graph for H, if G is H-free of maximum size, i.e. e(G) = ex(n; H). One of the earliest applications of extremal graph theory, by Erd}os,is to construct dense multiplicative Sidon set of integers using a graph without 4-cycles. The first result in extremal graph theory is the following theorem of Mantel, which answers Problem 1.1.1 when forbidding triangles as subgraphs. Theorem 1.1.2 (Mantel 1907). Let G be an n-vertex graph. If G is triangle-free, then 2 e(G) ≤ ex(n; K3) = bn =4c: 5 Exercise 1.1.3. Solve the puzzle at the beginning of this section, i.e. find the maximum 1 number of pairs of irrationals (xi; xj) with xi + xj being rational. Exercise 1.1.4. Prove that for any tree T , ex(n; T ) = O(n), that is, there exists a constant C = C(T ) such that ex(n; T ) ≤ Cn.2 Mantel's result in fact shows that extremal graph for triangle is Kbn=2c;dn=2e. This answers also the following natural question for triangles. Problem 1.1.5 (Extremal structure/Stability). How do H-extremal graphs look like? What about almost extremal graphs3, do they look like extremal ones? Theorem 1.1.2 was later generalised by Tur´anto forbidding larger cliques. To state his result, we need to define a special family of graphs. Let r 2 N, the r-partite Tur´angraph on n vertices, denoted by Tr(n), is the balanced complete r-partite n-vertex graph, i.e. each partite set is of size either bn=rc or dn=re. Clearly, Tr(n) is Kr+1-free. Theorem 1.1.6 (Tur´an1941). Let r ≥ 2 be an integer and G be an n-vertex graph. If G is Kr+1-free, then 1 n2 e(G) ≤ ex(n; K ) = e(T (n)) = 1 − − O(r): r+1 r r 2 Furthermore, the Tur´angraph Tr(n) is the unique extremal graph. We see from Tur´antheorem that there is a unique extremal graph Tr(n). The following theorem of Erd}osand Simonovits shows that this problem is stable in the sense that every almost extremal graph must be close in structure to the extremal Tur´angraph, answering Problem 1.1.5 for cliques. Theorem 1.1.7 (Erd}os-Simonovits stability 1966). Let " > 0, there exists δ > 0 such that the following holds. Let G be an n-vertex Kr+1-free graph. If 2 e(G) ≥ ex(n; Kr+1) − δn ; 2 then G can be changed to Tr(n) by altering at most "n adjacencies. 1.2 Zykov's symmetrisation There are many proofs for Tur´antheorem. Here we present one using Zykov's symmetri- sation. Zykov's symmetrisation is a process in which we alter the graph, one vertex at a time, 1Hint: Build an auxiliary graph and apply Mantel's theorem. 2Prove first that every graph with average degree d contains a subgraph with minimum degree at least d=2. 3We say G is almost extremal for H if G is H-free and close to maximum size, i.e. e(G) ≥ ex(n; H)−o(n2). 6 • without decreasing the number of edges, and • without increasing the clique number !(G).4 At the end of the process, we arrive to a complete partite graph, which has a much simpler structure to deal with. In particular, if the original graph is Kr+1-free, then all the graphs during symmetrisation will be Kr+1-free. Proof of Tur´antheorem via Zykov's Symmetrisation. Let G be an n-vertex Kr+1-extremal graph. Pick v1 2 V (G) with maximum degree and symmetrise all of its non-neighbours to v1. That is, for each u not adjacent to v1, set N(u) := N(v1). This operation keeps Kr+1-freeness and the resulting graph G1 has at least as many edges as G. Note that in G1, V1 := V n N(v1) is an independent set and completely joined to N(v1). We now repeat this operation as follows. Pick v2 2 G1[N(v1)] and symmetrise all its non- neighbours to v2. Let G2 be the resulting graph, then again G2 is Kr+1-free and e(G2) ≥ e(G1) ≥ e(G). Note that in G2, V2 := V (G2) n N(v2) is an independent set and completely joined to N(v2). 0 Continue this process, we will get a complete partite graph, say G , that is also Kr+1- free at the end. As the original graph G is an extremal Kr+1-free graph, together with e(G0) ≥ e(G), we see that G0 must be also extremal, i.e. e(G0) = e(G). We leave the uniqueness of extremal graph as exercise. Exercise 1.2.1. Among all Kr+1-free complete partite graphs, the Tur´angraph Tr(n) is the unique extremal graph. Further readings. Symmetrisation trick has been used in various extremal problems. To begin, one can read the linear algebraic version, and a recent generalisation due to F¨uredi and Maleki that can be applied to multiple graphs simultaneously. See also Pikhurko-Staden- Yilma for another application on Erd}os-Rothschild problem. • Motzkin and Straus, Maxima for graphs and a new proof of a theorem of Tur´an, Canad. J. Math., (1965). • F¨urediand Maleki, The minimum number of triangular edges and a symmetrization method for multiple graphs, Combin. Probab. Comput., (2017). • Pikhurko, Staden, and Yilma, The Erd}os-Rothschild problem on edge-colourings with forbidden monochromatic cliques, Math. Proc. Cambridge Phil. Soc. (2017). 1.3 Erd}os-Stonetheorem We have seen that Tur´antheorem determines the extremal number for cliques and describes the unique extremal structure. The natural next step is what if we forbid general graphs 4The clique number !(G) of a graph G is the order of the largest clique contained in G. 7 other than cliques? We shall present in this section a satisfying answer for all non-bipartite graphs. The seminal result of Erd}osand Stone shows that the extremal function of a general graph is completely determined by another important graph parameter: the chromatic number. Recall that the chromatic number of a graph H, denoted by χ(H), is the minimum number of colours needed to colour V (H) so that adjacent vertices do not receive the same colour. Theorem 1.3.1 (Erd}os-Stone1946). Let H be an arbitrary graph, then5 1 n2 ex(n; H) = 1 − + o(1) : χ(H) − 1 2 Note that by the definition of chromatic number, the (χ(H) − 1)-partite Tur´angraph is H-free, yielding the lower bound above.