1 - Introduction to hypergraphs

Transcription

1 1 - Introduction to hypergraph Jacque Vertraëte jacque@ucd.edu 1 Introduction In thi coure you will learn broad combinatorial method for addreing ome of the main problem in extremal combinatoric and then in other area of mathematic. Thi include the probabilitic method, pectral and polynomial method, and method from higher algebra. We begin with an introduction to hypergraph, which give a tate of different repreentation of hypergraph, linear hypergraph, and Turán-type problem, including exitence of Turán denitie and claification of zero Turán denitie. Thereafter we delve deeper into ome of the claical theorem of hypergraph theory, including variou theorem on interecting familie uch a Sperner Theorem, the LYM Inequality, the Erdő-Ko-Rado Theorem, Hilton-Milner Theorem, Deza-Frankl Theorem, Erdő-Rado Theorem and Frankl-Wilon Theorem. The tool include linear algebraic method, polynomial method, the delta-ytem method, compreion and hadow. The general Turán problem i conidered in the framework of analytic method and Lagrangian, and we conider ome pecific cae tudie where the exact anwer are known. We focu on extremal graph theory, where we preent Turán Theorem, the Erdő- Stone Theorem, the Kövari-Só-Turán Theorem, the Even Cycle Theorem and the Erdő-Gallai Theorem. We introduce the method of dependent random choice and we conider particular algebraic contruction for the bipartite Turán problem. The Combinatorial Nulltellenatz i introduced, and we tudy the problem of exitence of k-regular ubgraph in dene graph. 1

2 In the lat part of the coure we conider probabilitic and emirandom method, which include the Rödl Method in variou form with many application. A number of probabilitic tool which are ueful pecifically in extremal combinatoric i introduced. 1.1 Aymptotic notation Let f, g : N R + be function. Then we ue the following notation: f = O(g For ome contant c > 0, f(n cg(n for all n N. f = Ω(g f = o(g f = ω(g f = Θ(g f g f g g = O(f lim n f(n g(n = 0. g = o(f. f = O(g and f = Ω(g. lim n f(n g(n = 1. f(n lim up n g(n 1. If f = Θ(g, then we ay that f and g have the ame order of magnitude. 1.2 Real number inequalitie We frequently ue Jenen Inequality for convex function: if f : R R i convex and x 1, x 2,..., x n are real number and x i their average, then f(x 1 + f(x f(x n nf(x. Thi will often be tated a by convexity. We alo recall Taylor erie for tandard function uch a e x and log x, and in particular we have the ueful inequalitie: for 0 x < 1, log(1 x x and for x > 1, log(1 + x x. Thee lead to the inequality n n (1 + x i exp( x i for real x 1, x 2,..., x n > 1. We alo recall for x 1 and y 0, (1 + x y 1 + xy 2

3 and more generally if x = x n and y = y n and x n y n c, then (1 + x y e c. A particularly ueful formula to remember i Stirling Formula n! 2πn (n/e n, and more pecifically, n! = e θ n 2πn(n/e n 1 where 12n+1 θ n 1 12n. For non-negative real x i, y i, the Cauchy-Schwarz inequality i ued in the form ( n 2 n n x i y i x 2 i yj 2. Hölder Inequality, which generalize Cauchy-Schwarz, tate that if 1 p + 1 q = 1 where p, q 1, then j=1 n ( n 1/p ( n 1/q. x i y i x p i y q i Any further inequalitie required will be developed locally. 1.3 Baic definition and hypergraph notation Let 2 V denote the power et of a et V and let ( V k denote the ubet of V of ize k. A hypergraph i a pair (V, E where V i a et and E 2 V. The element of V are called vertice and the element of E are called edge. A multihypergraph i a pair (V, E where V i a et and E i a multiet of ubet of V in other word we allow repeated edge. If H i a hypergraph, we write V (H for the et of vertice and E(H for the et of edge. If E ( V k then H i called k-uniform and H i called a k-graph. We often write only E for a hypergraph (V, E with the undertanding that V (H = e E e and write e(h intead of E(H. The denity of a k-graph H on n vertice i e(h/ ( n k. For a et S V (H, let N H (S = e\s denote the neighborhood of S and e H:e S d H (S = {e E(H : e S} denote the degree of S. When there i no ambiguity we uppre the ubcript H. Alo note that for 2-graph H, d H (v = N H (v for every vertex v V (H. A hypergraph H i d-regular if for every v V (H, d H (v = d. An iolated vertex i a vertex of degree zero. 3

4 Propoition 1. If H i a hypergraph, then ( e = r e E(H S V (H: S =r d H (S. In particular, if H i an n-vertex d-regular k-graph, then e(h = dn/k. Propoition 1 generalize the handhaking lemma for graph, which i the cae k = 2 and r = 1. It i convenient to let Kt k be the complete k-graph on t-vertice the edge et i with V = t. Let Kt:k denote the complete k-partite k-graph with part of ( V k ize t, namely, V (K t:k = V 1 V 2 V k where V 1 = V 2 = = V k and E(K t:k = {{v 1, v 2,..., v k } : v 1 V 1, v 2 V 2,..., v k V k }. For k = 2 and k = 3 we may ometime write K t,t and K t,t,t for the complete k-partite k-graph. If S i a et of vertice in a hypergraph H, then we denote by H S the hypergraph with vertex et V (H\S and edge et {e H : e S = }. If E i a et of edge in H, let H E denote the hypergraph with vertex et V (H and edge et E(H\E. If S V (H, then H[S] i the hypergraph (S, {e H : e S}: thi i the ubgraph induced by S. The hadow hypergraph of a hypergraph H, denote H, i the hypergraph with vertex et V (H and edge et {e {x} : e E(H, x e}. We denote by k H the kth hadow hypergraph, namely, ( (... ( H.... In many intance, one can obtain information on H from H, and many claical theorem on interection in hypergraph are naturally proved uing hadow. The link hypergraph of a et S V (H i the hypergraph H S with vertex et V (H\S and edge et {e\s : e E(H, S e}. 1.4 Repreentation of hypergraph In the lat ection we defined a hypergraph to be a pair (V, E where E i a family of ubet of E. There are other ueful repreentation of hypergraph, each of which we viit briefly below. Incluion repreentation. The bigraph repreentation of a hypergraph H = (V, E i a bipartite graph with part V and E, where v V i joined to e E if v E. Thi allow one ometime to ue graph theory on the bigraph repreentation to deduce fact about the original hypergraph. In general, one can conider any 4

5 incluion repreentation, whereby we create a bipartite graph with part ( V r and E and where S ( V r i joined to e E if S e. It i ometime convenient to conider the incluion matrix of the hypergraph, which i exactly the incidence matrix of the incluion repreentation. In other word, if H i a hypergraph, we can form the matrix I whoe row are indexed by ( V r and whoe column are indexed by E, where I S,e = 1 if e S and I S,e = 0 otherwie. Thi i ometime ueful to bring the tool of linear algebra on I S,e to give information about H. Tenor repreentation. In graph theory, thi i often done via the adjacency matrix of the graph, and the natural generalization of thi to k-graph i the adjacency tenor. Namely, given a k-graph H = (V, E, form the n n n n tenor A indexed by V V V where A v1,v 2,...,v k = 1 if {v 1, v 2,..., v k } E(H and A v1,v 2,...,v k = 0 otherwie. It i poible to introduce ome linear algebra here via multilinear form, but the theory i rather pare, and there doe not eem to be a ueful notion of rank and pectrum of tenor for combinatorial purpoe. Graph from hypergraph. If H i a k-graph, then we could pick number r, uch that r+ = k, and we could form a graph whoe vertex et i ( V r ( V coniting of edge {R, S} where R ( V r and S ( V and R S = e for ome e E(H. When r, thi graph i bipartite, but when r =, thi i not necearily bipartite. A well-known graph i the Kneer graph K n k : given n and k, the vertex et of K n k i ( V k where V = n, and the edge et of Kn k i the et of dijoint pair of element of ( V k. In other word, two k-element ubet of V are adjacent if they are dijoint. Then a k-graph H = (V, E can be viewed a a ubgraph of K n k. If, for intance, every two edge of H interect, then H correpond to an independent et in K n k. Thi i one of the key connection in the proof of the Erdő-Ko-Rado Theorem. Duality. Finally, we mention a natural notion of duality in hypergraph. Given a multihypergraph H = (V, E, we can form a dual multihypergraph H = (E, V where the element of V are indexed by the vertice v V and where the edge e v indexed by v i preciely {e E : v e}. For intance, if H i a d-regular k-uniform hypergraph, then H i a k-regular d-uniform multihypergraph. In the event that H i a 2-regular hypergraph, H i a multigraph, and we can bring in the tool of graph theory. Other repreentation. In application the hypergraph that arie have are ometime contructed from algebraic, geometric or group theoretic ource. For intance, 5

6 given an abelian group Γ and a et S Γ, one may define the notion of a Cayley k- graph H with vertex et Γ and edge et {{x 1, x 2, x 3,..., x k } Γ : x 1 +x 2 + +x k S}. Another example i to take a vertice the 1-dimenional ubpace of an n- dimenional vector pace over F q, and the edge a the et of 1-dimenional ubpace which form k-dimenional ubpace. Thi lead to the claical contruction of projective plane when n = 3 and k = 2. For a geometric example, one could take a vertex et the toroidal grid T n = Z n Z n, and a edge et all triple of the form {(x, y, (x + a, y, (x, y + b} B where a 0 and b 0 and addition i modulo n. Thee kind of contruction will come up fairly frequently in extremal problem, and are often worthy of tudy in their own right. 1.5 Turán-type problem for hypergraph A hypergraph F i a ubgraph of hypergraph H if V (F V (H and E(F E(H, and we write F H. If F i a et of hypergraph, we ay that H i F-free if F H for all F F. The central problem of extremal combinatoric i to determine or etimate We alo let ex(n, F := max{e(h : H 2 [n] and H i F-free}. ex k (n, F = max{e(h : H ( [n] and H i F-free}. k Thee problem are collectively referred to a Turán-type problem. In thi coure we focu mainly on the econd problem. Any F-free k-graph with n vertice and ex k (n, F edge i called an extremal hypergraph for F. Propoition 2. For any F, the following limit exit ex k (n, F lim ( n n. k Proof. The exitence of the limit follow from the fact that π n (F = ex k (n, F/ ( n k i non-increaing a a function of n and bounded below by 0. To ee thi, count pair (e, T where T i a et of n 1 vertice in an F-free n-vertex hypergraph H, and e T i an edge of H. The number of pair i at mot n ex k (n 1, F. On the other hand, the number of pair i exactly ( n k = e(h(n k. n 1 k e E(H 6

7 Therefore Taking limit we get ex k (n 1, F (n k ( n 1 e(h k n ( n 1 = e(h ( n. k k π n 1 (F π n (F. The limit i denoted π(f, the Turán denity of F. The above proof ha a number of powerful conequence. The firt i the Simonovit uperaturation theorem: Propoition 3. Fix k 2 and let F be a t-vertex k-graph. For every ε > 0, there exit δ > 0 uch that if H i any n-vertex k-graph with ex k (n, F + εn k edge, then H contain δ ( n t copie of F. Thi propoition ay that the copie of F aturate the moment a k-graph pae above the denity π(f. A conequence of the proof of Propoition 2 i that π n (Kt k π t (Kt k = 1 ( t 1, k and therefore: Corollary 4. For all n, π(k k t 1 In general, thi corollary i not tight. ( t 1. k An analytic approach of Sidorenko uing Feynman integral greatly improve the upper bound; we hall come to thi topic later. In the cae of graph, we hall ee that one can determine π(f exactly for any F, via the Erdő-Stone Theorem. For hypergraph, very little i known. Let u conider a cae tudy. Simple bound for π(k4 3 are a follow: Propoition π(k Proof. Firt we give the upper bound. Let H be an n-vertex K4 3 -free hypergraph. We oberve that ince any four vertice in the hypergraph contain at mot three edge. S V (H S =2 ( ( d(s n By convexity, and Propoition 1, the um on the left i at leat ( 3e(H/N N 2 7

8 where N = ( ( n 2. A calculation give e(h 2 1 n 3, a required. For the lower bound, a contruction of a dene K4 3 -free hypergraph i required. Form an n-vertex hypergraph H = (V, E where V = X Y Z and X Y Z X + 1, and where E conit of all edge {x, y, z} with x X, y Y and z Z, and all edge {a, b, c} where a, b X and c Y, or a, b Y and c Z, or a, b Z and c X. The total number of edge i ( ( X Y X Y Z + Y Dividing by ( n 3 give π(k Z + ( Z X 5n Equality hold in the upper bound of Propoition 5 only if almot all quadruple carry exactly three triple. However, for any x, y V (H, there are ( n d(x,y 2 2 quadruple containing S which carry at mot two triple. Thi idea allow the bound in Propoition 5 to be improved ubtantially. We do not invetigate the bet bound on π(k k t at thi tage, except to ay that the tate of the art i to conidering larger ubet of V (H, one ue the method of flag algebra to generate inequalitie which give bound on π(k4 3 which are much cloe to 5 9. However, Turán conjecture remain open: Conjecture 6. π(k 3 4 = 5 9. One difficulty with Turán conjecture i that there are infinitely many aymptotically extremal non-iomorphic hypergraph, a contructed by Kotochka, other than the one given in the proof above. The ituation for Kt k i apparently even more difficult, and Erdő offer one thouand dollar for the determination of π(kt k for any pair (t, k with t > k > Degenerate Turán Problem In preceding ection, we aw that determining π(f for general familie of hypergraph F i an open quetion. However, a pecial cae i to determine thoe familie F for which π(f = 0. Thee are called Degenerate Turán Problem. An intereting feature of thee problem i that the extremal hypergraph tend to be random-like (in a ene that will be made precie, and the contruction tend to have very rich geometric or algebraic tructure. 8

9 1.6.1 k-partite k-graph We aim to how that k-partite k-graph are thoe k-graph reponible for a family F having zero Turán denity. A k-graph H i aid to be k-partite if V (H admit a partition V 1 V 2 V k uch that for every e E(H and every i k, e V i = 1. The et V i are called the part of H. If H i any k-graph, then a cut of H i a k-partite ubgraph containing all vertice of H. A key ueful reult due to Erdő and Kleitman how that every k-graph ha a relatively dene k-partite ubgraph: Propoition 7. If H i a k-graph, then H ha cut with at leat k! k k e(h edge. Proof. We randomly, uniformly and independently aign a color c(v from {1, 2,..., k} to each vertex v of H. In other word, for 1 i k, P(c(v = i = 1/k. Then fixing an edge e E(H, the probability that all the vertice in e have different color (we ay e i multicolored i k!/k k. In particular, the expected number of multicolored edge of H i k! k k e(h by linearity of expectation. Pick an intance of a coloring of V (H uch that at leat k!e(h/k k edge of H are multicolored, and let V i be the et of vertice that received color i for i k. Then the k-graph H of all multicolored edge i the required cut. If we would like the part in the cut to have a cloe to the ame ize a poible, then a lightly different proof can be ued. A partition V 1 V 2 V k of a et V i called balanced if V 1 V 2 V k V in other word, the part are a equal in ize a poible. A balanced cut of a k-graph H i a cut whoe part form a balanced partition of V (H. A hint of the proof of the following i given. Propoition 8. If H i a k-graph, then H ha a balanced cut with at leat k! k k e(h edge. Proof. We count in two way pair (e, P uch that P = V 1 V 2 V k i a balanced partition of V (H and e V i = 1 for 1 i k. The detail are left a an exercie. We leave the reader to think about an algorithm which efficiently find, given a k-graph H, a balanced cut with at leat k! k k e(h edge. 9

10 1.6.2 Kövari-Só-Turán Theorem for graph We return now to the zero Turán denity quetion. To tart gently, we firt conider the cae of graph: for which familie F of graph i π(f = 0? It turn out that the bipartite graph are reponible for thi. To prove thi, we ue the Kövari-Só-Turán Theorem: Theorem 9. For all t 2, ex 2 (n, K,t (t 1 1/ n 2 1/ + ( 1n. Proof. Let H be a K,t -free graph with n vertice. We claim ( ( dh (v n (t 1. v V (H To ee thi, note that the left hand ide count pair (v, S where v V (H and S N H (v ha ize. If there are more than (t 1 ( n uch pair, then ome fixed et S V (H of ize mut have appeared in more than t 1 pair (v, S. In other word, there mut be vertice v 1, v 2,..., v t which all have S in their neighborhood. Thi i preciely a copy of K,t with part S and T = {v 1, v 2,..., v t }. By convexity and Propoition 1, v V (H ( ( dh (v d n where d = 2e(H/n i the average degree of H. Therefore ( ( d n n (t 1 and thi how Thi implie n! (d + 1 (t 1 n!. d (t 1 1/ n 1 1/ + 1 and ince d = 2e(H/n we find e(h (t 1 1/ n 2 1/ + ( 1n. Therefore π n (K,t = O(n 1/ and taking limit π(k,t = 0 a required. 10

11 From thi it quickly follow that π(f = 0 if and only if F contain a bipartite graph. Corollary 10. A family F of graph ha π(f = 0 if and only if F contain a bipartite graph. Proof. If F contain no bipartite graph, then the complete bipartite graph K a,n a with part of ize a = n/2 and n a are F-free and aymptotically have denity 1 2. Therefore π(f 1 2 in thi cae. If F contain a bipartite graph F, then F i contained in K,t for ome, t. By Theorem 9, Therefore π(f = 0. ex 2 (n, F ex 2 (n, K,t (t 1 1/ n 2 1/ + ( 1n. In fact from Theorem 9 we deduce that for any finite family F containing a bipartite graph, there exit γ > 0 uch that ex 2 (n, F = O(n 2 γ, o the denity zero reult i in a very trong ene. One may ak for the behavior of ex 2 (n, F for other bipartite graph, however thi i known a the bipartite Turán problem and i generally very difficult. Erdő and Simonovit made a number of related conjecture, perhap the mot important being the exponent conjecture: Conjecture 11. For every finite family F containing a bipartite graph F, there exit an exponent α [1, 2 uch that ex 2 (n, F = Θ(n α. Thi conjecture i wide open for mot clae of graph, and i not known for uch imple graph a octagon, cube and K 4,4. We will return in depth to the exponent quetion when we introduce more advanced counting method and contruction. We hall alo ee that there are infinite familie of graph with no exponent. In the cae of k-graph with k > 2, the ituation i even more complicated. We hall ee that Conjecture 11 cannot be extended to k-graph: there are finite familie F of k-graph with no exponent k-partite k-graph have zero Turán denity We claimed that if F i a family of k-graph then π(f = 0 if and only if F contain a k-partite k-graph. A key propoition i the following: 11

12 Propoition 12. Let d 1 be a real number and k > r 1. If H i an n-vertex k-graph with (d 1 ( n r edge, then H ha a ubgraph H uch that the degree of every r-et in H i at leat d. We leave the proof a an exercie. We return now to how π(f = 0 when F contain a k-partite k-graph. It i ufficient to how π(k t:k = 0 for every t N. Proceed by induction on k, noting that when k = 2 thi follow from the Kövari- Só-Turán Theorem. Suppoe π(k t:k 1 = 0 and let H be an n-vertex k-graph with εn k edge. By Propoition 12, H ha a ubgraph J with minimum degree at leat εn k 1. Suppoe V (J = m. Then each link hypergraph J v i a (k 1-graph with at leat εn k 1 εm k 1 edge. Since π(k t:k 1 = 0, the number of copie of K t:k 1 in J v i at leat δn (k 1t for ome δ > 0 by the uperaturation lemma. Therefore the number of pair (v, K with K a copy of K t:k 1 in J v i at leat δm (k 1t+1. Thi how that there i ome et X of (k 1t vertice uch that there are at leat δm pair (v, K where V (K = X. There are T = ( t(k 1 t,t,t,...,t way to partition X into et of ize t, and o there are δm/t pair (v, K where K i ome fixed K t:k 1. For large enough m, δm/t t, which mean we get t identical copie of K t:k 1 which are in the link graph J v1, J v2,..., J vt for ome vertice v 1, v 2,..., v t V (H. Thi implie H contain all edge of the form {v i } e where e K t:k 1, which i preciely K t:k on the vertex et {v 1, v 2,..., v t } V (K t:k 1. In fact, thi counting proof can be ubtantially refined to give an upper bound on ex k (n, K t:k of the form O(n k 1/tk 1, akin to the Kövari-Só-Turán Theorem when k = Kövari-Só-Turán Theorem for hypergraph The Kövari-Só-Turán Theorem allow u to actually count copie of K,t in a dene graph. The following propoition i due to Erdő and Moon: Propoition 13. Let G be an n-vertex graph of denity p uch that ( n e(g = p / n 2 1/ + 2n. Then the number of copie of K, in G i at leat Ω(p 2 n 2. 12

13 Proof. Firt we note that if M = v V (G ( dg (v then uing the lower bound on e(g and convexity, ( 2e(G/n M n n. Now the number of copie of K, in a graph G i exactly 1 2 ( f(s S V (G where the um i over et S of ize, and f(s i the number of vertice which are adjacent to every vertex in S. Since M n ( n, ( ( ( ( f(s n M/ n = Ω(n 2 M. S V (G Since M = Ω(e(G n 1, n 2 M = Ω(p 2 n 2, a required. We now ue thi to how that the familie of 3-graph F uch that π(f = 0 are preciely the 3-partite 3-graph. Thi follow from the following theorem. Theorem 14. For any t 2, ex k (n, K t:k = O(n k 1/tk 1. Proof. We prove the reult for k = 3 and leave the generalization to k > 3 a an exercie. Let H be a K t,t,t -free n-vertex 3-graph. Suppoe e(h = ω(n 3 1/t2. By Propoition 12, there i a ubgraph J of H of minimum degree d e(h/n. Let m be the number of vertice of J. The number of edge in the link graph J v i exactly the degree of v in J, o we conclude e(j v e(h/n = ω(m 2 1/t2 for every v V (J. By Propoition 13, J v contain ω(m 2t 1 copie of K t,t. We therefore have ω(m 2t pair (v, K where v V (J and K = K t,t J v. It follow that for ome fixed K = K t,t, there are ω(1 pair (v, K, and in particular, there are vertice v 1, v 2,..., v t V (J uch that K J vi for all i t. Then {v 1, v 2,..., v t } V (K i the vertex et of K t,t,t J H, contradicting that H i K t,t,t -free. Therefore e(h = O(n 3 1/t2. 13