LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS. Complete bipartite graph.. Kövari-Só-Turán theorem. When H i a bipartite graph, i.e., when χ(h) = 2, Erdő and Stone theorem aert that π H = 0. In other word, for each fixed poitive real ε, there exit n 0 uch that for n > n 0, if G i an n-vertex graph with at leat εn 2 edge, then G contain H a a ubgraph. Thu bipartite graph are degenerate cae in extremal problem for graph. We prove the following much tronger reult, due eentially to Kővari, Só, and Turán. (Note that for every bipartite graph H, there exit and t uch that H K,t ). Theorem. (Kővari, Só, Turán) For any natural number and t with t, there exit a contant c uch that ex(n, K,t ) cn 2 /. Proof. Suppoe that G i an n-vertex graph with at leat cn 2 / edge. Count the pair (v, S) coniting of a vertex v V (G) and a et S N(v) of ize S =. The number of uch pair i ( ) d(v) n( n v v d(v) ) ( ) 2cn / n n c n! = c n! for n ufficiently large. If c > t, then there exit a et S that i counted at leat t time in the equation above. Thi give a copy of K,t in G. A more careful analyi how that the following bound hold: () ex(n, K,t ) ( + o()) 2 (t )/ n 2 /. It i known that thi bound i harp in ome pecific cae. We will come back to thi later. We preent a quick application of Kővari-Só-Turán theorem to the unit ditance problem of Erdő. Theorem 2. A et of n point in R 2 determine at mot cn 3/2 unit ditance. Proof. Let X be a given et of n point in R 2. Let G be a graph on the vertex et X where two point x, y are adjacent if and only if x y =. Note that G i K 2,3 -free. Therefore by Kővari-Só-Turán theorem, G ha at mot cn 3/2 edge.

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 2 The bet known bound to the problem i O(n 4/3 ) firt proven by Szemerédi and Trotter. By now there are everal different proof of the ame bound. Any improvement on the exponent would be extremely intereting. It i conjectured that the bound hould be n +o(). The contruction that motivated thi bound i the n n grid. If we re-cale the grid o that the mot popular ditance i, then there are about n +c/ log log n unit ditance in thi contruction..2. Random Contruction. The binomial random graph G(n, p) i the probability pace of graph with vertex et [n] where each pair of vertice form an edge independently with probability p. We ay that G(n, p) ha a property P aymptotically almot urely if the probability that G(n, p) ha P tend to a n tend to infinity. Here we examine the value of p for which G(n, p) contain K,. Let N be the random variable counting the number of copie of K,. For a fixed pair of et X, Y of ize X = Y =, the probability that the bipartite graph between X and Y i complete i p 2. Therefore by linearity of expectation, we have E[N] n2 2(!) 2 p2. If n 2 p 2 = (n 2 p ) < 2 (thi happen when p n 2/ ), then ince > E[N] P(N ), 2 we know that N = 0 with probability at leat 2. Furthermore, it i well known that in thi range of p, the number of edge in G(n, p) i a.a.. ( n 2) p n 2 2/. One can lightly improve thi contruction. Note that if N n2 p 4, then we can remove one edge from each copy of K, to obtain a K, -free graph on n vertice with at leat (about) n2 p 4 edge. Note that E[N] < n2 p 4 occur when p n 2/(+). Therefore for all 2, c n 2 2/(+) ex(n, K, ) c 2 n 2 /..3. Algebraic Contruction. The following elegant contruction, firt dicovered by Erdő, Rényi, and Só in 966, how that the bound (??) i tight for = t = 2 for infinitely many value of n. For a prime power q, we denote the finite field of order q by F q. Theorem 3. (Erdő, Rényi, and Só) For infinitely many value of n, there exit a graph on n vertice and about n 3/2 edge where each pair of vertice have exactly one common neighbor. Proof. Let p be a prime. Define a equivalence relation over F 3 p where two point (a, a 2, a 3 ), (a, a 2, a 3 ) are equivalent if and only if there exit λ F p uch that a i = λa i for all i =, 2, 3. For a point (a, b, c) F 3 p, let [a, b, c] be

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 3 the equivalence cla that contain (a, b, c) (thi pace i called a projective plane). Let V = {[a, b, c] : a, b, c F p, (a, b, c) (0, 0, 0)} and note that V = p 3 p = p2 + p +. Let G be a graph over vertex et V where [a, b, c] and [a, b, c ] are adjacent if and only if aa + bb + cc = 0. We claim that each pair of vertice in G have exactly one common neighbor. Indeed, let v = [a, b, c] and v = [a, b, c ] be two ditinct vertice of G. Then [x, y, z] can be adjacent to both vertice only if ax + by + cz = 0 and a x + b y + c z = 0. Since [a, b, c] and [a, b, c ] are ditinct vertice, the et of vector (x, y, z) F 3 p atifying both equation above form a line in F 3 p. Thi mean that the non-zero olution define a ingle equivalence cla. Hence there exit a unique common neighbor of v and v. For the diagonal cae = t, the only other cae where (??) i known to give the correct order of magnitude for ex(n, K, ) i when = t = 3. Brown conidered the graph over F 3 p where two vertice (a, b, c) and (a, b, c ) are adjacent if and only if (a a ) 2 + (b b ) 2 + (c c ) 2 =, and howed that thi i K 3,3 -free. For 4, the diagonal cae i not known but when t i much larger than, the bound i known to be tight up to the contant. Thi wa firt proved by Kollár, Rónyai, and Szabó for t! +, and wa then improved by Alon, Rónyai, and Szabó to t ( )! +. All the above mentioned contruction can be put into the following framework. Let V and V 2 be two copie of F k p and f be ome polynomial over F p in 2k variable. Each cae wa baed on ome clever polynomial f. Recently, Bukh howed ome intereting family of contruction by conidering random polynomial f intead. 2. Tree In thi ection, we invetigate the Turán number of tree and even cycle. The following lemma i another extremely ueful tool in extremal graph theory. Lemma 4. Let G be an n-vertex graph with at leat nd edge. Then there exit a ubgraph of minimum degree at leat d +. Proof. Repeatedly remove vertice of degree at mot d from the graph. If we top before and there exit at leat 2 vertice, then we found a ubgraph of minimum degree at leat d +. Otherwie, we removed n vertice and there are no remaining edge. Therefore e(g) (n )d and thi contradict the given condition. Let T be a tree on t + vertice. One can eaily prove that t n ex(n, T ) (t )n 2

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 4 where the lower bound i for n diviible by t. The lower bound follow from conidering a graph coniting of vertex dijoint copie of K t. The upper bound follow from Lemma?? ince if G i an n-vertex graph with at leat (t )n edge, then it contain a graph of minimum degree at leat t into which we can greedily embed any tree on t + vertice. Hence the extremal number for tree i linear in the number of vertice. A famou conjecture of Erdő and Só aert that the lower bound i the right anwer for all tree T. Recently, Ajtai, Komló, Simonovit, and Szemeredi announced a poitive olution to thi conjecture for ufficiently large n. 3. Even cycle The following bound on the extremal number for even cycle wa proved by Bondy and Simonovit. Theorem 5. For each fixed poitive integer t 2, there exit a contant c uch that ex(n, C 2t ) cn + t. By Lemma??, we can find a ubgraph G of minimum degree at leat cn /t. Fix an arbitrary vertex v of G and conider a breadth-firt-earch tree T contructed by tarting the exploration at v. For each l [0, t], let L l be the et of vertice at the l-th level of T. If all we wanted wa a cycle of length at mot 2t (intead of exactly 2t), then we can eaily finih the proof ince for each l t, the inequality L l+ cn /t L l hold, howing that L t c t n, and for c contradict the fact that G ha at mot n vertice. However, it require a coniderable amount of work to find a cycle of length exactly 2t. A imilar phenomenon occur frequently in the tudy of extremal number of bipartite graph, where one can eaily prove the exitence of a non-injective copy but i a lot more difficult to find an injective copy. Theorem?? i known to be the correct order of magnitude only for t = 2, 3, 5. Note that for t = 2, C 4 i the ame a K 2,2. Hence the contruction given above how the tightne for t = 2. For t = 3 and 5, the matching lower bound were found by Benon, and Singleton repectively. Recently, uing the framework developed by Bukh mentioned in a previou ection, Conlon made an intereting breakthrough. He howed that for every t, there exit an uch that for infinitely many value of n, there exit an n-vertex graph in which there are at mot path of length t between each pair of vertice. 4. Spare bipartite graph The following theorem extend Kövari-Só-Turán theorem. Theorem 6. (Füredi) There exit a contant c uch that the following hold. Let H be a bipartite graph with bipartition A B where each vertex a A ha at mot d neighbor in B. Then for ufficiently large n, ex(n, H) cn 2 /d.

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 5 We prove thi theorem uing dependent random choice. Lemma 7. Suppoe that G i a graph on n vertice with at leat αn 2 edge. Then for any integer r, there exit a ubet of vertice A of ize A 2 αr n uch that all r-tuple of vertice in A have at leat 2 αn/r common neighbor. Proof. Let v,, v r be vertice choen uniformly and independently at random, and define A = r i= N(v i). Note that x ( A if and ) only if v,, v r are all choen from N(a). Hence P(x A) = N(x) r. n Therefore by the linearity of expectation and convexity, E[ A ] = ( ) N(x) r nα r. n x V (G) Let M be the number of r-tuple of vertice in A with le than 2 αn/r common neighbor. ( Note) that for a fixed r-tuple of vertice R, the probability that R A i d(r) r, n where d(r) i the number of common neighbor of R. Therefore by linearity of expectation, ( ) r E[M] nr αn /r = αr r! 2 r!2 r n. Therefore E[ A M] nαr 2. There exit a particular choice of v,, v r, for which A M nαr 2. For thi choice, let A be the et obtained from A by removing one vertex from each r-tuple with le than 2 αn/r common neighbor. Note that A A M nαr 2, and all r-tuple of vertice in A have at leat 2 αn/r common neighbor. The proof of Theorem?? eaily follow. Proof of Theorem??. Suppoe that H i a bipartite graph with bipartition X Y, where X Y = h and all vertice in X have degree at mot d. Let G be graph with n vertice and at leat 2hn 2 /d edge. Apply Lemma?? with α = 2hn /d and r = d to find a et A of ize A 2 cd h uch that all d-tuple of vertice in A have at leat 2c h common neighbor. We can now eaily find a copy of H by a greedy algorithm. Firt arbitrarily embed X into A and call thi map f. For each vertex y Y, note that f(n(y)) ha at leat h common neighbor. Since X Y = h, we can find one vertex in the common neighborhood that i not an image of f. Define f(y) a thi vertex and repeat. In the end, we obtain a copy of H. A graph i d-degnerate if all it ubgraph have a vertex of degree at mot d. Degeneracy i a natural meaure of parene of a graph. For example if a graph i d-degenerate then every ubet of vertice X contain at mot X d edge. Alo, there exit a linear ordering of the vertice o that each

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 6 vertex ha at mot d neighbor that precede itelf. Erdő made the following conjecture in 967. Conjecture 8. (Erdő) For every d-degenerate bipartite graph H, ex(n, H) = O(n 2 /d ). The bet known reult toward thi conjecture i the following theorem of Alon, Krivelevich, and Sudakov. It proof i alo baed on dependent random choice. Theorem 9. (Alon-Krivelevich-Sudakov) There exit a contant c uch that the following hold. If H i a d-degenerate bipartite graph then for ufficiently large n, ex(n, H) cn 2 /(4d). At one point Erdő made the following conjecture which eem plauible given all the reult mentioned above. Conjecture 0. (Erdő) For any graph bipartite graph H, there exit α = ex(n,h) α(h) uch that lim n n exit. Furthermore, perhap α = + α k or α = 2 k. The econd part wa diproved by Erdő and Simonovit. They howed (among other reult) that there i a graph whoe extremal number i Ω(n 2 8/7 ) and O(n 8/5 ). (the bipartite join of Θ(3, 3) and K 2 where Θ(3, 3) i two vertice connected by three internally dijoint path of length 3). There i alo the following conjecture. For a family of graph L, define ex(n, L) a the maximum number of edge in an n-vertex graph which contain no ubgraph from the family L. Conjecture. (Erdő-Simonovit) For every rational α (0, ), there exit a finite family L for which ex(n, L) = Θ(n +α ). Some intereting familie of bipartite graph L are minor and topological ubdiviion of complete graph. We will not addre thee topic in the coure but intereted reader can refer to Dietel graph theory book. 5. Erdő-Simonovit and Sidorenko conjecture Erdő and Simonovit conjecture addree an intereting quetion regarding multiplicity of bipartite graph. Conjecture 2. (Erdő-Simonovit) For every bipartite graph H, there exit contant γ (0, ), c > 0, and n 0 uch that graph G on n n 0 vertice with m n 2 γ edge contain at leat cn V (H) p E(H) copie of H where p = ( m i the edge denity of G. n 2) Note that the expected number of copie of H in a random graph G(n, p) i roughly Aut(H) n V (H) p E(H). Therefore the conjecture above in ome

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 7 ene aert that random graph contain (or, i cloe to containing) the minimum number of copie of H. Sidorenko made a imilar conjecture on multiplicity of homomorphim of bipartite graph. A homomorphim of a graph H to G i a map f : V (H) V (G) for which {f(v), f(w)} i and edge whenever {v, w} i an edge. Note that the map need not be injective. For two graph H and G, let Hom(H, G) be the et of all homomorphim from H to G, and let t H (G) = Hom(H,G) V (G) V (H) be homomorphim denity. We may interpret t H (G) a the probability that a uniform random map from H to G form a homomorphim. A beautiful conjecture of Sidorenko aert that Conjecture 3. For all bipartite graph H and all graph G, t H (G) t K2 (G) E(H). Thi i an amazingly elegant conjecture; it doe not require G to be a large graph and there i no contant involved. The conjecture fail for all non-bipartite graph. For example if H i K 3, then we can take G = K 2 for which the left-hand-ide i 0 and the right-hand-ide i 8. When H i a bipartite graph conider the cae when G i the random graph G(n, p). Then by linearity of expectation, we have Hom(H, G) = (+o())n V (H) p E(H) (the o() term i there to account for the non-injective copie). Therefore t H (G) = (+o())p E(H). For H = K 2, it give t K2 (G) = ( + o())p. Thu the inequality above i almot tight for random graph. Sidorenko conjecture aert that for every bipartite graph H, the number of homomorphim from H to G over n vertex graph of fixed denity i aymptotically minimized when G i random-like (thi can be made formal but we will not go into detail at thi point). Note that the cae when H i a path i already non-trivial. Surpriingly, the conjecture turn out to be equivalent to Erdő and Simonovit conjecture. One direction i eay. Sidorenko implie Erdő-Simonovit. Given a bipartite graph H, take γ = E(H). Then the number of non-injective homomorphim from H to G i at mot n V (H). On the other hand by Sidorenko conjecture, the number of homomorphim from H to G i at leat n V (H) p E(H). Since p > 2n γ (approximately), n V (H) p E(H) n V (H) 2 n V (H) p E(H). Therefore the number of copie of H in G i at leat 2 Aut(H) n V (H) p E(H). Erdő-Simonovit implie Sidorenko. Lemma 4. For all G and N, there exit a graph G on at leat N vertice for which t H (G ) = t H (G) for all graph H.

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 8 Proof. Let G be a blow-up of G obtained by replacing each vertex with N vertice. Let π : V (G ) V (G) be the canonical projection homomorphim. Note that h : V (H) V (G ) i a homomorphim if and only if π h : V (H) V (G) i a homomorphim. Therefore and t H (G ) = Hom(H, G ) V (G ) V (H) = Hom(H, G ) = N V (H) Hom(H, G) Hom(H, G ) Hom(H, G) N V (H) V (G) V (H) = V (G) V (H) = t H (G). Suppoe that Erdő-Simonovit conjecture i true but Sidorenko conjecture i not true. Let (H, G) be a pair that diprove Sidorenko conjecture. Thu H i a bipartite graph and G i a n-vertex graph with p = t K2 (G) and t H (G) = ( ε)t K2 (G) E(H) where ε > 0. The following lemma i known a the tenor power trick. Lemma 5. For all G and k, there exit a graph G for which t H (G ) = t H (G) k for all graph H. Proof. Let G G be a graph with vertex et V (G) V (G) where two vertice (v, w) and (v, w ) are adjacent if and only if {v, v } and {w, w } are both edge in G. We claim that graph H, Hom(H, G G) = Hom(H, G) Hom(H, G) where the correpondence i given by h Hom(H, G G) mapping to π h and π 2 h for the projection map π and π 2 from V (G) V (G) to the firt coordinate, and econd coordinate, repectively. Therefore Hom(H, G) 2 t H (G G) = V (G) 2 V (H) = t H (G) 2. The product G G i known a the tenor product of G and G. One can imilarly define tenor product for different graph. The concluion follow by conidering the tenor product of G with itelf k time. Let c be the contant coming from Erdő-Simonovit conjecture for H and define k a the minimum integer for which ( ε) k < c. By Lemma?? there exit a graph G for which t H (G) = ( ε) k t K2 (G) E(H) < ct K2 (G) E(H) = cp E(H). On the other hand by Lemma?? we may aume that the number of vertice of G i ufficiently large in term of the edge denity of G. We mut then have Hom(H, G ) c(n ) V (H) p E(H). Thi i a contradiction ince Hom(H, G ) = t H (G ) (n ) V (H). We eentially proved that Sidorenko conjecture i equivalent to the following conjecture.

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 9 Conjecture 6. Let H be a bipartite graph. There exit a poitive contant c uch that for every p (0, ), every large enough graph G with n vertice and at leat 2 n2 p edge contain at leat cn V (H) p E(H) copie of H. Sidorenko conjecture i known to hold for few familie of graph: path, tree, even cycle, complete graph, hypercube, etc. We ay that a bipartite graph H ha Sidorenko property if the conjecture hold for H. 5.. Path. Conider a path H = P k where k 2 (path with k edge). We prove that Hom(H, G) 2 2k n k+ p k hold for all n-vertex graph with at leat 2 n2 p edge. We will prove the tatement by induction on n. If n =, then p = 0 and the claim trivially hold. Let G be an n-vertex graph with at leat 2 n2 p edge and uppoe that the claim ha been proved for all maller value of n. Suppoe that G ha αn vertice of degree le than 4 np. Let G be the graph obtained by removing all vertice of degree le than 4np. Note that G ha ( α)n > 0 vertice and at leat 2 n2 p αn 4 np = 2 ( α/2)n2 p edge. Therefore if α > 0, then by the inductive hypothei, Hom(H, G) Hom(H, G ) ( ) α/2 k 2 2k+ ( α)k+ n k+ ( α) 2 p = 2 2k+ ( α/2) k ( α) k nk+ p k 2 2k+ nk+ p k. where the final inequality follow ince k 2. Hence we may aume that α = 0, i.e., that G ha minimum degree at leat 4 np. In thi cae we can contruct a homomorphim from P k+ to G by arbitrarily chooing it firt vertex and greedily extending it k time. Thi way each homomorphim i counted twice and therefore the number of path of length k i at leat 2 n ( 4 np ) k 2 2k+ nk+ p k. Therefore Sidorenko conjecture hold for path. 5.2. Complete bipartite graph. Let H = K t, for ome fixed integer t and. Let G be an n-vertex graph with m = 2 n2 p edge. The number of pair (v, S) with S = and S N(v) i M = v V (G) ( ) ( ) deg(v) np/2 n = ( + o()) 2! n+ p where the lat inequality hold for large enough n depending on p (in other word K, atifie Sidorenko conjecture). For a et S, let d(s) be the

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS 0 number of common neighbor of S and note that M = S V (G), S = d(s). Note that ) ) ( np /2 S V (G) ( d(s) t ( ) n ( ( M/ n t ) n! t ) nt+!t!2 t pt. Thi prove that complete bipartite graph atify Sidorenko conjecture. 5.3. Graph with a complete vertex. Conlon, Fox, and Sudakov proved that if H i a bipartite graph with two part A B where there exit a vertex a 0 A adjacent to all other vertice in B. Let a = A and b = B. Let ε be a mall enough poitive real. For each integer k [b], we call a k-tuple of vertice X good if X ha at leat ε 2 np k common neighbor. Call it bad otherwie. Define V k V a the et of vertice v for which N(v) contain at leat ε deg(v) k bad k-tuple. Note that n k (ε 2 np k ) ε k deg(v) k ε V k deg(v). V k v V k v V k Since V k n, the above implie that v V k deg(v) ε /k n 2 p. Define V 0 = V V b. Let U be the et of vertice having degree at leat εnp. Note that if ε i mall enough, then deg(v) 2 n2 p εnp deg(v) 4 n2 p. v U\V 0 v V 0 v V \U Take a vertex v U \ V 0. We ay that a b-tuple of vertice X in N(v) i excellent if every ubet Y X i good. By definition, the number of excellent b-tuple in N(v) i at leat b ( ) b deg(v) b ε deg(v) k deg(v) b k k 2 deg(v)b. k= Conider a homomorphim from H to G where we map a 0 A to v and map B to ome excellent b-tuple. The number of way to embed a 0 and B a above i at leat 2 deg(v)b 2 n deg(v) n v U\V 0 v U\V 0 b 2 n 4 b nb p b = 2 2b+ nb+ p b. For each a A other than a 0, note that the image of N(a) i ome good deg(a)-tuple and hence ha at leat ε 2 np deg(a) common neighbor. We may chooe the image of a to be any one of uch vertice. Therefore the number of

LECTURE 2. TURÁN S THEOREM : BIPARTITE GRAPHS way to extend the partial homomorphim above to a full homomorphim i at leat ε 2 np deg(a) ε 2(a ) n a p e(h) b. a A\{a 0 } Hence the total number of homomorphim of H to G i at leat 2 2b+ nb+ p b ε 2(a ) n a p e(h) b ε2(a ) 2 2b+ na+b p e(h). Thi prove that Sidorenko conjecture hold for H. Thi approach ha been further generalized by Szegedy-Li, Kim-Lee-Lee, and Szegedy. The mallet graph for which Sidorenko conjecture i not known to be true i K 5,5 \ C 0. Reference [] M. Aigner, Turán Graph Theorem [2] N. Alon, J. Spencer, Probabilitic Method. [3] D. Conlon, Extremal graph theory lecture note (Lecture ). [4] R. Honberger, Mathematical Diamond [5] Z. Füredi and M. Simonovit, The hitory of degenerate (bipartite) extremal problem.