Lecture 11: Extremal graph theory II October 13, 2020 () Lecture 11 October 13, 2020 1 Review Recall that if H is a (finite, simple) graph, we denote by ex(n; H) the maximum number of edges of a (simple) graph G on n vertices that does not contain any subgraph isomorphic to H. Note: we always assume that E(H) 6= ;. The Erd¨os-Stonetheorem implies that: ex(n;H) 1 • If χ(H) ≥ 3, then limn!1 n2 = 1 − χ(H)−1 > 0. • If χ(H) = 2 (equivalently, H is bipartite), then for every > 0, we have ex(n; H) ≤ n2 for n 0. In this lecture we begin by discussing the behavior of ex(n; H) for n ! 1 for certain examples of bipartite graphs H. Trivial example. If H is the graph below and if G does not contain H, 1 then deg(v) ≤ 1 for every v 2 V (G). Hence #E(G) ≤ 2 #V (G). () Lecture 11 October 13, 2020 2 The case of P4 The first interesting case of a bipartite graph is that of H = P4. Theorem 1. If G is a simple graph on n vertices that does not contain any P , then 4 n p #E(G) ≤ (1 + 4n − 3): 4 Proof. For every v 2 G, let's put dv = deg(v). We consider F = f(u; v; w) j fu; vg 2 E(G); fu; wg 2 E(G); v 6= wg : We count #F in two ways: • Every u 2 V (G) contributes du(du − 1) to F , hence P #F = u2V (G) du(du − 1). • For every pair of distinct vertices (v; w), there is at most one (u; v; w) 2 F since G contains no P4, hence #F ≤ n(n − 1). () Lecture 11 October 13, 2020 3 The case of P4, cont'd We thus conclude that X X 2 X n(n − 1) ≥ du(du − 1) = du − du: (1) u2V (G) u2V (G) u2V (G) P Recall now that if m = #E(G), then u du = 2m. We thus have 0 12 X 2 X 2 n n(n − 1) + 2m ≥ n · du ≥ @ duA = 4m ; u2V (G) u2V (G) where the first inequality follows from (1) and the second one from Cauchy-Schwarz. This gives 4m2 − 2mn − n2(n − 1) ≤ 0 and by computing the roots of this quadratic expression, we get n p m ≤ 4 (1 + 4n − 3). () Lecture 11 October 13, 2020 4 Sharpness of the estimate for ex(n; P4) Remark. The result in Theorem 1 is sharp in the sense that there is c > 0 3=2 such that ex(n; P4) > cn for infinitely many n. In order to see this, consider n = p2, where p is a prime number and V = Z=pZ × Z=pZ; hence n := #V = p2. Two distinct (x; y) and (x0; y 0) in V are joined by an edge if x + x0 = yy 0. Note that for every (x; y) 2 V there are precisely p pairs (x0; y 0) 2 V with x + x0 = yy 0. This is clear if y 6= 0 (we can take x0 arbitrary and then y 0 = y −1(x + x0)); on the other hand, if y = 0, then x0 = −x and y 0 can be arbitrary. However, it might happen that (x0; y 0) = (x; y). In any case, it follows that for every v 2 V , we have deg(v) 2 fp − 1; pg, hence 1 X p2(p − 1) #E(G) = deg(v) ≥ > cn3=2 2 2 v for some c > 0 and all p 0. () Lecture 11 October 13, 2020 5 Sharpness of the estimate for ex(n; P4), cont'd Claim: for every p, the graph we have described contains no subgraph isomorphic to P4. Indeed, suppose that two distinct (x1; y1) and (x2; y2) 2 V have two distinct neighbors (x; y) and (x0; y 0). Since x + x1 = yy1 and x + x2 = yy2; it follows that x1 − x2 = y(y1 − y2): If y1 = y2, this implies x1 = x2, a contradiction. On the other hand, if y1 6= y2, we see that y is determined uniquely by −1 y = (x1 − x2)(y1 − y2) , and then also x is determined uniquely by x = yy1 − x1. We thus again obtain a contradiction. 3=2 2 Conclusion: we have ex(n; P4) > cn for n = p 0. () Lecture 11 October 13, 2020 6 The complete bipartite graphs Recall that the complete bipartite graph Km;n is the bipartite graph on two sets A and B with m and n elements, respectively, and such that every element of A is a neighbor of every element of B. Note that P4 is isomorphic to the complete bipartite graph K2;2. Our next goal is to extend (partially) the result of Theorem 1 to all graphs Kn;n. Theorem 2. For every t ≥ 2, there is c > 0 such that 2− 1 ex(n; Kt;t ) < c · n t for all n 0: Remark. Of course, if this is the case, then after possibly replacing c by a larger constant, we may assume that the inequality holds for all n. () Lecture 11 October 13, 2020 7 The complete bipartite graphs, cont'd Proof of Theorem 2. Suppose that G is a graph on f1;:::; ng that does not contain any subgraph isomorphic to Kt;t . Put di = deg(i). Note that di the set consisting of the neighbors of i has precisely t subsets with t elements. Since G contains no Kt;t , when we vary i, every such subset appears ≤ (t − 1) times. We thus have n t X di n n ≤ (t − 1) ≤ (t − 1) : (2) t t t! i=1 Note that d1 + ::: + dn = 2e, where e = #E(G). Since the function x x(x−1):::(x−t+1) f (x) = t = t! is convex on [t; 1), it follows that we have the following lower bound for the left-hand side of (2): n 2e 2e 2e t X di 2e=n n · ::: − t + 1 n · − t + 1 ≥ n = n n ≥ n : t t t! t! i=1 () Lecture 11 October 13, 2020 8 The complete bipartite graphs, cont'd By combining the inequalities on the previous slide, we obtain 2e t n · − t + 1 ≤ (t − 1)nt : n Taking t-roots, we get 2e 1=t 1− 1 ≤ (t − 1) n t + t − 1; n and thus 2− 1 e < c · n t for a suitable c > 0 and all n 0. () Lecture 11 October 13, 2020 9 Polygons with an even number of vertices One can extend the result in Theorem 1 in another direction, by considering polygons with an even number of vertices. We state the following result, without proof: Theorem 3. There is c > 0 such that 1+ 1 ex(P2k ; n) ≤ c · n k for all n: () Lecture 11 October 13, 2020 10 Edge density We end our introduction to extremal combinatorics with the presentation of a powerful result, due to Szemer´edi,which over the years found many striking applications. We will only discuss the statement and an outline of the proof, and describe briefly some applications. In what follows we only consider finite simple graphs. Szmer´edi's Regularity Lemma roughly says that for a graph with many vertices, the vertex set admits a partition into a bounded number of subsets such that the edges between most different parts behave \random-like". Definition. Given a graph G with vertex set V and two (nonempty) subsets X ; Y ⊆ E(G), we put EG (X ; Y ) = f(x; y) 2 X × Y j fx; yg 2 E(G)g and the edge density between X and Y is given by jE (X ; Y )j d (X ; Y ) = G 2 [0; 1]: G jX j · jY j () Lecture 11 October 13, 2020 11 -regular pairs and -regular partitions We fix a finite simple graph G with vertex set V and > 0. We now introduce a condition on two subsets X ; Y of V , that roughly says that for two large subsets of X and Y , the edge density between these subsets approximates the edge density between X and Y . Definition. A pair (X ; Y ) of nonempty subsets of V is -regular if for every subsets A ⊆ X and B ⊆ Y , with jAj ≥ jX j and jBj ≥ jY j, we have jdG (A; B) − dG (X ; Y )j ≤ . We next consider partitions of V . Ideally, we would want partitions such that any pair of sets is -regular. However, we have to settle for less: most of such pairs will be regular, with the precise meaning of \most" depending on the size of the sets. More precisely, we make the following definition. () Lecture 11 October 13, 2020 12 -regular pairs and -regular partitions Definition. A partition P of V given by fV1;:::; Vr g is an -regular partition if X 2 jVi j · jVj j ≤ jV j : 1≤i;j≤r (Vi ;Vj )6=−regular With this definition, we can now state Theorem 4 (Szemer´edy'sRegularity Lemma). For every > 0 and every positive integer m, there is an integer M ≥ m such that for every graph G on a set of vertices V , with jV j ≥ M, there is an -regular partition of V into k sets, with m ≤ k ≤ M. () Lecture 11 October 13, 2020 13 Endre Szemer´edi Endre Szemer´edi,Hungarian mathematician, Professor at Rutgers University He is famous (among other things) for proving a conjecture of Erd¨osand Tur´anon sequences of positive integers with positive density.