A Hypergraph Extension. Bipartite Turán Problem

Transcription

1 A Hypergraph Extesio of the Bipartite Turá Problem Dhruv Mubayi 1 Jacques Verstraëte Abstract. Let t, be itegers with t. For t, we prove that i ay family of at least t 4( ) triples from a -elemet set X, there exist t triples A 1, B 1, A, B,..., A t, B t ad distict elemets a, b X such that A i A j = {a} ad B i B j = {b}, for all i j, ad { A i {a} = B j {b} for i = j A i B j = for i j. Whe t =, we improve the upper boud t 4( ) to ( ) + 6. This improves upo the previous best kow boud of.5 ( ) due to Füredi. Some results cocerig more geeral cofiguratios of triples are also preseted. 1 Itroductio Let F be a family of r-graphs, some member of which is r-partite. A fudametal theorem due to Erdős states that there exists δ = δ(f) > 0 such that the maximum umber of edges i a r-graph o vertices cotaiig o member of F is O( r δ ) as. The asymptotic order of this maximum, deoted ex(, F), is geerally very difficult to determie. For surveys, we refer the reader to Füredi [6] ad to Frakl [8]. I this paper, we cosider the above problem for the followig specific classes of r-graphs. 1 Departmet of Mathematics, Statistics, ad Computer Sciece, 851 S. Morga Street, Chicago, IL mubayi@math.uic.edu Faculty of Mathematics, Uiversity of Waterloo, 00 Uiversity Aveue West, Waterloo, ON, Caada NL G1 jverstraete@math.uwaterloo.ca This research was doe while both authors were postdocs at Microsoft Research. 000 AMS Subject Classificatio: 05B05, 05B07, 05C5, 05C65, 05D05. Keywords: triple system, r-graph, Steier system, bipartite Turá problem.

2 Defiitio. Let X 1, X,..., X t be t pairwise disjoit sets of size r 1, ad let Y be a set of s elemets, disjoit from i [t] X i. The K (r) s,t deotes the r-graph with vertex set i [t] X i Y ad edge set {X i {y} : i [t], y Y }. Remark. Note that K (r) s,t ad K (r) t,s are oisomorphic whe r ad s t. Our results apply to both cases, so for simplicity throughout this paper we let t s. Defiitio. Let f r () be the maximum umber of edges i a vertex r-graph cotaiig o four edges A, B, C, D with A B = C D ad A B = C D =. I the case r =, we ote that f () = ex(, K (),). Erdős [] asked whether f r () = O( r 1 ) whe r. Erdős ad Frakl (upublished) proved that f r () = O( r 1/ ). Füredi [4] later aswered Erdős questio by the followig Theorem: Theorem 1.1 (Füredi) For all itegers, r with r ad r, ( ) 1 + r 1 ( ) 1 f r () <.5. r r 1 The lower boud arises from the family of all r-elemet subsets of [] cotaiig a fixed elemet of [] together with a arbitrary family of 1 pairwise disjoit r-elemet r subsets ot cotaiig that elemet. Füredi also observed that if we replace every 5-set i a Steier S 1 (, 5, ) family by all its -elemet subsets, the the resultig triple system has ( ) () triples ad cotais o copy of K, (for the existece of S 1 (, 5, ), see Ray- Chaudhuri ad Wilso [1]). This slightly improves the lower boud above whe r = to ( ). Füredi cojectured that this costructio gives a sharp lower boud whe 1, 5 modulo 0, ad that the lower boud i Theorem 1.1 is sharp for r 4 ad sufficietly large. I this paper, we will cocetrate o triple systems excludig a copy of K (),t ad, more geerally, excludig a copy of K () s,t. This is a commo geeralizatio of the problem of estimatig both f () = ex(, K,) () ad ex(, K () s,t ). The latter is a fudametal ope problem i extremal graph theory. Our mai result also improves Füredi s upper boud for f r () i Theorem

3 Theorem 1. Let t ad t be itegers. The ex(, K () ( ) + 6 for t =,t ) < t 4( ) for t >. Moreover, for ifiitely may, ex(, K (),t ) t 1 ( ). Remark: The expressio ex(, K (),t )/ ( ) has a limit g(t) as. This follows by similar argumets to Propositio 6.1 i [4]. By Theorem 1., g(t) t 4. It would be iterestig to determie the growth rate of g(t). Usig Theorem 1. ad Lemma 5. i [4] oe ca easily obtai the followig improvemet to Theorem 1.1 (see the remark at the ed of Sectio 4). Corollary 1. Fix r. The f r () < ( ) r 1 + O( r ). We will prove the lower boud of Theorem 1. i Sectio. I Sectio, we prove a fudametal lemma which eables us to establish the upper boud of Theorem 1. i Sectio 4. Geeralizig Theorem 1. to the larger class K () s,t for s > seems more difficult, ad we are ot able to determie the order of magitude of ex(, K () s,t ). We prove the followig extesio of the result of Erdős ad Frakl that f () = ex(, K,) () = O( 5/ ). Theorem 1.4 Let / t s. The where c s,t depeds oly o s ad t. ex(, K () s,t ) < c s,t 1/s, (1) The proof of Theorem 1.4 (see Sectio 5) follows easily from the well-kow boud ex(, K s,t ) < c s,t 1/s (see [10]) for graphs. However, we believe that the expoet 1/s i (1) is ot the truth for ay s. Theorem 1. shows this for s =. As further evidece, we ca also improve (1) whe s = (see Sectio 6). This proof does ot yield improvemets for larger values of s. 5

4 Theorem 1.5 Let t. The ex(, K (),t ) < c t 1/5, where c t depeds oly o t. I the other directio, we prove the followig Theorem i Sectio. Theorem 1.6 ex(, K () [ex(, K s,t )] s,t ) > d s,t. It is believed (see Füredi [6]) that ex(, K () s,t ) = Ω t ( 1/s ) wheever s t, but this has oly bee proved for s ad t > (s 1)! (see Kollár, Róyai, Szabó [9] ad Alo, Róyai, Szabó [1]). This immediately implies the followig Corollary to Theorem 1.6. Corollary 1.7 If t > (s 1)! > 0, the ex(, K () s,t ) > d t /s. We feel that /s is the correct order of magitude of ex(, K () s,t ). Cojecture 1.8 Let s, t be itegers with s t. The ex(, K () s,t ) = Θ t ( /s ). Notatios. The symbol [] deotes the set {1,,..., }, ad X (r) deotes the family of all r-sets i set X. We write G for a simple fiite udirected graph, uless idicated otherwise, ad G(A, B) to idicate that G is bipartite with parts A ad B. The otatio Γ G (v) is used for the set of vertices adjacet to a vertex v i a graph (or hypergraph) G, e(g) is the umber of edges i G, deg G (v) is the umber of edges icidet with vertex v i G, ad (G) is the umber of vertices i G. For ay vertex u of G, we write G u for the subgraph of G spaed by all edges of G disjoit from u. Similarly, if E is a set of edges of G, the G E deotes the subgraph of G spaed by all edges of G which are ot i E. A hypergraph cotaiig o subgraph isomorphic to a fixed hypergraph F is called F-free. 6

5 Lower Bouds for ex(, K () s,t ) I this sectio, we give lower bouds for the umbers ex(, K () s,t ), by showig that they are related to the bipartite Turá umbers ex(, K s,t ) = ex(, K () s,t ). We will establish this relatioship usig the followig costructio: Costructio. Let G be ay K () s,t -free bipartite graph, with parts A = {a 1, a,..., a } ad B of size. Let A = {a 1, a,..., a }. Defie a -partite triple system H o A B A whose triples cosist of those sets (a i, b, a j) for which a i ba j is a path i G. We write z(, K (r) s,t ) for the maximum umber of edges i a r-partite K (r) s,t -free r-graph i which all parts have size. Propositio.1 The triple system H cotais o K () s,t ad z(, K () s,t ) e(h) 1 z(, K s,t) z(, K s,t ). Proof. Choose G i the above costructio to cotai z(, K () s,t ) edges. The umber of triples i H is precisely twice the umber of paths a i ba j i G. Therefore, by the covexity of biomial coefficiets, e(h) = v B ( ) degg (v) ( ) e(g)/ = (e(g) / e(g)/) = e(g) / e(g). Thus H has the required umber of edges. We ow check that H is K () s,t -free. Suppose, for a cotradictio, that F H is isomorphic to K () s,t. We suppose the edges of F are {X i {y} : i [t], y Y }, where Y is a s-elemet set, X 1, X,..., X t are t pairwise disjoit -elemet sets, ad Y is disjoit from i [t] X i. It is ot hard to see that there exists a -partitio of F ito parts Z 1, Z ad Z, each of which is cotaied etirely i a distict part of H. We suppose Z 1 = s ad Z = Z = t. If Z 1 A (Z 1 B), the we may assume Z B (Z A) by symmetry. By the costructio, Z 1 Z iduces a copy of K () s,t i G, a cotradictio. Therefore Z 1 = {a 1, a,..., a s} A. By symmetry, we assume Z B. Let Z = {a 1, a,..., a s }. The Z Z iduces a copy of K () s,t i G, a cotradictio. This completes the proof of Propositio.1. 7

6 The above propositio immediately gives the followig result, by otig the K () s,t -free orm graph costructios of Kollár, Róyai ad Szabó [9] for t > (s 1)! (see also Alo, Róyai ad Szabó [1]), ad the costructios due to Füredi [7] of K,t -free graphs o vertices with at least (t 1) 1/ / c 4/ edges, where c is a positive costat: Corollary. Let s, t be itegers with s ad t > (s 1)!. The z(, K () s,t ) = Ω t ( /s ). Moreover, for all t, z(, K (),t ) (t 1) d 11/6 for some costat d > 0. Propositio.1 also establishes the lower boud i Theorem 1.6, sice ex(, K () s,t ) z(, K () s,t ) ad z(, K s,t ) ex(, K s,t ), where the last iequality ca be foud i []. Füredi s costructio for ex(, K (),) ca easily be geeralized for ex(, K (),t ), thereby improvig Propositio.1 i the case s =. Ideed, cosider a S 1 (, t + 1, ) Steier system (i.e. every pair of elemets is cotaied i precisely oe (t + 1)-set) i which we replace each (t + 1)-set by all its -elemet subsets. The existece of such Steier systems is established i Ray-Chaudhuri ad Wilso [1] wheever = t modulo t(t 1). The resultig triple system is K (),t -free ad the umber of triples is ( )( ) 1 ( ) t + 1 t + 1 This verifies the lower boud i Theorem 1.. = t 1 ( ). Mai Lemma I this sectio, we establish a geeralizatio of a lemma due to Füredi (see Lemma. i [4]). This eables us to give a upper boud o the umber of edges which may be deleted from a graph to obtai a K s,t -free subgraph. This lemma will be of fudametal importace i the proofs of both Theorem 1. ad Theorem 1.5. We begi with the followig defiitio: Defiitio. Let t s, ad let G = G(A, B) be a bipartite graph. The D s,t (G) deotes the s-graph o A B whose edge set cosists of those S A (s) B (s) for which S lies i a K s,t i G. 8

7 Mai Lemma. Let G = G(A, B) be a bipartite graph ad let t s. The we may delete at most (s + t )e(d s,t (G)) edges from G to obtai a K s,t -free graph. Proof. We proceed by iductio o e(g). For coveiece, we write D istead of D s,t (G) ad D E istead of D s,t (G E). If e(g) = 0, the e(d) = 0. We also suppose G has o isolated vertices. Suppose e(g) > 0. If some edge f G is i o K s,t i G the, by iductio, we may remove at most (s + t )e(d f ) = (s + t )e(d) edges from G f to delete all K s,t i G f, ad hece all K s,t i G. We therefore assume every edge of G is cotaied i a K s,t i G. We ow aim to defie a o-empty set E of edges of G such that E (s + t )[e(d) e(d E )]. Let us see that this will suffice to complete the proof. The iductio hypothesis will apply to G E: we delete a set E of at most (s + t )e(d E ) edges from G E to obtai a K s,t -free graph. The total umber of edges deleted is E + E (s + t )[e(d) e(d E )] + (s + t )e(d E ) = (s + t )e(d). For a cotradictio, we suppose that o such set E exists. That is, for ay o-empty set E of edges of G, E > (s + t )[e(d) e(d E )]. ( ) Claim 1. deg D (u) < deg G (u) for every vertex u G. Proof. Suppose deg G (u) deg D (u) for some u G. Let E be the set of edges of G icidet with u. The we certaily have 0 < E = deg G (u) deg D (u) (s + t )deg D (u), as s, t. Sice o K s,t i G E cotais u, u is a isolated vertex of D E. This implies deg D (u) e(d) e(d E ). Cosequetly, E (s+t )deg D (u) (s+t )[e(d) e(d E )], cotradictig ( ). So deg D (u) < deg G (u) for every vertex u G. Fix a vertex u A, ad let B = Γ G (u). Let A 1,..., A k deote the edges of D icidet with u. Let B i = x A i Γ G (x) for i [k], ad defie A = {A i : B i t 1} ad B to be the hypergraph spaed by the edge set {B i : B i t}. Note that B i B for all i as u A i, ad B may have multiple edges. Claim. A =. Proof. Suppose A. Let E i be the set of edges from u to B i, for each i [k], ad set E = A i A E i. Sice B i for each i, we have E. O the other had E E i = B i (t 1) A, A i A A i A 9

8 where the last iequality follows from the defiitio of A. Suppose A i A lies i a K s,t i G E. The B i cotais the part of this K s,t i B. However, all edges betwee u ad B i lie i E, so these edges are abset i G E. This implies o A i A is a edge i D E, ad hece E (t 1)[e(D) e(d E )]. This cotradicts ( ). Claim. There exists v B such that {B i : v B i } Γ B (v). Proof. As each edge of G is i some K s,t i G ad A =, B = i [k] B i. By Claim 1, (B) = B = deg G (u) > deg D (u) = e(b). Applyig Propositio A.1 to B, there exists a vertex v B such that {B i : v B i } = deg B (v) < Γ B (v) + 1. This completes the proof of Claim. We ow defie E = {vx : x u ad vx lies i a K s,t i G with at least s vertices i B }. Let D be the subgraph of D iduced by B, ad let D E be the subgraph of D E iduced by B. If some edge S of D E is icidet with v, the there exists a K s,t i G E cotaiig S. As this K s,t cotais s vertices i B, all its edges (apart from uv) icidet with v are i E. This cotradictio shows v is a isolated vertex i D E, ad deg D (v) e(d) e(d E ). It remais to verify that E (s + t )deg D (v) for E to cotradict ( ). This fact is established i the ext two claims. For each G(A i, B i ) with v B i, select a sigle edge betwee v ad A i, distict from uv. Let the collectio of selected edges be E, ad set E = E E. Claim 4. E (s 1)deg D (v). Proof. Sice A =, every set A i has at least t commo eighbors i B. This implies that B i iduces a complete s-graph i D, for i [k]. Therefore deg D (v) (B i {v}) (s 1). i:v B i By Lemma A., applied to the hypergraph B with edge set i:v B i (B i {v}) (s 1), we fid Γ B (v) = (B i {v}) (s 1) (B i {v}) (s 1) (s 1)deg D (v). i:v B i i:v B i 10

9 Cosequetly, by Claim ad the defiitio of E, E {B i : v B i } Γ B (v) (s 1)deg D (v). This completes the proof of Claim 4. Claim 5. E (t )deg D (v). Proof. Suppose E > (t )deg D (v). The there exists a edge S of D icidet with v ad a set T of t 1 vertices of A {u} icidet with all of S. However, u is also adjacet to all vertices of S so G(S, T {u}) is a K s,t cotaiig u. Cosequetly, vx E for some vertex x T. This cotradicts E E =, sice we also have vx E for every x T. This completes the proof of Claim 5. We have show that E = E + E (s + t )deg D (v) (s + t )[e(d) e(d E )], cotradictig ( ). This completes the proof of the Mai Lemma. 4 Upper Bouds for ex(, K (),t ) I this sectio, we establish the upper bouds i Theorem 1.. For itegers q, s, t, we write K () q,s,t for the complete -partite -graph with parts of sizes q, s, t. Our proof of Theorem 1. will use the coutig techique from Mubayi [11] together with the Mai Lemma i Sectio. This Mai Lemma allows us to remove a small umber of triples that destroy all copies of K () 1,,t i a K (),t -free triple system. A additioal refiemet of these ideas allows us to prove that ex(, K,) () < ( ) + 6. It is sufficiet to restrict our attetio to -partite triple systems, i view of the followig useful lemma of Erdős ad Kleitma [5]: Lemma 4.1 Let G be a triple system o vertices. The G cotais a -partite triple system, with all parts of size, ad with at least e(g) triples. 9 Ideed, we prove the followig theorem: Theorem 4. Let t, t, ad g(t) = t 1/ + (t 1) [( ) ] + 1. The z(, K (),t ) < for t = g(t) for t >. 11

10 Let us verify Theorem 1. from Theorem 4.. By addig at most two isolated vertices to a K () s,t -free triple system H o vertices, we obtai a triple system G such that (G) is divisible by three. Applyig Lemma 4.1, we fid a -partite triple system G with at least e(g) edges. By Theorem 4. ad Lemma 4.1, 9 9 e(h) = 9 e(g) e(g ) < g(t) ( (G) ) = 1 9 g(t)(g). Cosequetly, e(h) = e(g) 9 [ ] 1 9 g(t)(g) 1 ( ) g(t)( + ) < t 4. The last iequality follows by some elemetary calculatios, usig t ad t. A similar argumet applies to show that ex(, K,) () < ( ) + 6. Before provig Theorem 4., we require the followig defiitio: Defiitio. Let G 1, G,..., G be graphs o the same vertex set. The i [] G i deotes the multigraph i which a pair f of vertices is a edge wheever f is a edge of some G i. Proof of Theorem 4.. Let H be a -partite K (),t -free triple system, i which all three parts have size. Suppose the parts of H are each copies of [], labeled A, B, C. For each i [], let G i = G i (A, B) deote the bipartite graph with edge set { (a, b) : a A, b B, (a, b, i) H }. Let G = i [] G i, D(A) = i [] D(G i ) A ad D(B) = i [] D(G i ) B, where D(G i ) A deotes the subgraph of D(G i ) = D,t (G i ) iduced by A, ad similarly for B. Claim 1. A pair of vertices {a, a } i A () B () forms a part of a copy of K,t i G i for at most t 1 itegers i []. Proof. Suppose some pair {a, a } A () forms a part of a copy of K,t i G i for at least t itegers i [], say for i [t]. The there exist t vertices b 1, b,..., b t B such that ab i a is a path of legth two i G i. However, the set of all edges of the form (a, b i, i) ad (a, b i, i) forms a copy of K (),t i H. This is a cotradictio, so {a, a } forms a part of a copy of K,t i G i for at most t 1 itegers i []. Claim. For t, D(A) ad D(B) have edge-multiplicity at most () ( ) +(). 1

11 Proof. Suppose, for a cotradictio, that some edge {a, a } has edge multiplicity at least r = (t 1) ( ) + t i D(A). Without loss of geerality, we may assume {a, a } is a edge of D(G i ) A for each i [r]. By Claim 1, {a, a } is cotaied i a t-set formig a part of a K,t i G i for at least r t + 1 itegers i [r], say for i [r t + 1]. This gives a set of r t + 1 edges i D(B), correspodig to the part of a K,t i G i of size two, for i [r t + 1]. These r t + 1 edges together spa a multigraph M D(B). The pairs of adjacet vertices i M form a part of a K,t i differet graphs G i, for i [r t + 1]. Now e(m) r t + 1 = (t 1) ( ) + 1 ad M has edge-multiplicity at most t 1 by Claim 1. So we may apply Lemma A.4 (with s = ad µ = t 1) to M: there exist t vertices b 1, b,..., b t B, each icidet with a differet edge of M. Suppose this set of edges is f 1, f,..., f t with f i D(G i ) for i [t]. The ab i a is a path of legth two i G i for i [t]. But the, for each i [t], all the edges (a, b i, i) ad (a, b i, i) form a copy of K (),t i H. This cotradictio verifies Claim for D(A). Similar argumets apply to show that D(B) has edge-multiplicity at most (t 1) ( ) + (t 1). For each i [], let G i = G i(a, C) deote the bipartite graph spaed by the edges (a, c), a A, c C, for which (a, i, c) is a edge of H. Let G = i [] G i, D (A) = i [] D(G i) A ad D (C) = i [] D(G i) C. We ote, by symmetry ad applyig the argumets of Claim, that D (A) ad D (C) have edge-multiplicity at most (t 1) ( ) + (t 1) for t. Claim. We may remove at most 4(t 1) [ ( ) ( ) + 1] edges from G G so that, for all i [], the resultig subgraph of G G cotais o K,t i ay G i or G i. If t =, we may remove at most ( ) edges from G G for the same coclusios. Proof. Suppose t. By Claim, o pair of vertices of A or B is a edge i more tha (t 1) ( ) + (t 1) of the graphs D(Gi ). By the Mai Lemma, we may delete at most (t 1)e(D(G i )) edges from each G i to obtai a K,t -free graph. The umber of edges removed from G is therefore at most (t 1)e(D(A)) + (t 1)e(D(B)) [ ( ) t 1 (t 1) + (t 1) ] ( A similar argumet applies for G, therefore the total umber of edges removed is at most 4 [ (t 1) ( t 1 ) + (t 1) ] ( ). ). 1

12 Now suppose t =. We assert that D(A) D (A) = (this is the major ew idea eeded to improve the factor i Füredi s boud from.5 to ). This suffices to prove Claim : as D(A) has o multiple edges by Claim 1, the Mai Lemma shows that the umber of edges required to delete all K, i G i ad G i is at most ( ) ( ) ( ) ( ) A B C e(d(a)) + e(d (A)) + e(d(b)) + e(d (C)) + + =. Let us prove D(A) D (A) =. Suppose, for a cotradictio, that {a, a } is a edge of D(G i ) ad D(G j), where a, a A. The there are edges {b, b } D(G i ) ad {c, c } D(G j) such that {a, b, a, b } ad {a, c, a, c } iduce quadrilaterals i G i ad G j respectively. Now (a, b, i), (a, b, i), (a, b, i), (a, b, i) ad (a, j, c), (a, j, c ), (a, j, c ), (a, j, c) are all edges of H. These edges form a triple system cotaiig a copy of K (),, a cotradictio. This completes the proof of Claim. We let H G ad H G deote the subgraphs of G ad G obtaied by removig all these edges from G ad G. For vertices x, y i a hypergraph, the codegree of x ad y, writte codeg(x, y) is the umber of edges cotaiig both x ad y. Claim 4. e(h G ) < (t 1/). Proof. Suppose e(h G ) (t 1/). The the umber of paths with two vertices i A ad oe vertex i B, cotaied i both H G ad some G i, is exactly ( ) codeg(b,c) (b,c) B C. As the average codegree is at least t 1/, the above expressio is miimized whe the codegree of half the pairs is t 1, ad the codegree of the other half of the pairs is t. Therefore (b,c) B C ( ) codeg(b, c) 1 ( ) t ( ) ( ) t > (t 1). This implies the existece of a set P B C of (t 1) + 1 pairs ad a, a A such that the triples (a, b, c) ad (a, b, c) are edges of H wheever (b, c) P. Let G deote the bipartite graph o B C whose edges are the elemets of P. By Lemma A., G cotais a matchig M with t edges or a star with t edges. I the former case, the set all of triples of the form (a, b, c) ad of the form (a, b, c), with {b, c} M, form a copy of K (),t i H, a cotradictio. I the latter case, we obtai a K,t i G i or G j, accordig as the ceter 14

13 of the star is j B, or i C. As H G cotais o K,t i ay G i, by Claim, this is a cotradictio. So e(h G ) < (t 1/), ad the proof of Claim 4 is complete. We ow complete the proof of Theorem 4.. First suppose t. Recall that G = G i ad H G is the subgraph of G remaiig o deletig edges from G usig Claim. Let D deote the umber of edges deleted i Claim. Thus, usig Claims ad 4, e(h) e(h G ) + D [( ) ] ( ) t 1 < (t 1/) + 4(t 1) + 1 ( [( ) ]) t 1 < t 1/ + (t 1) + 1. For t =, by Claims ad 4, we fid e(h) < ( ) +. Remark: Corollary 1. ca be proved i the same way as Theorem 4., usig the geeralizatio of Lemma 4.1 to r-partite subgraphs of r-uiform hypergraphs, due to Erdős ad Kleitma [5], ad usig Lemma 5. i [4]. 5 Upper Boud for ex(, K () s,t ) I this sectio we prove Theorem 1.4. Proof of Theorem 1.4: It suffices to prove that z(, K () s,t ) < c s,t 1/s. Let A, B, C be the three parts of size of a -partite K () s,t -free triple-system H. Suppose that H has more tha c s,t 1/s triples, where c s,t is defied as the smallest iteger for which every bipartite graph with parts X ad Y of size with more tha c s,t 1/s edges cotais a K s,t with t vertices i X ad s vertices i Y. Note that c s,t is idepedet of, sice the umber of edges betwee X ad Y must satisfy ( ) ( ) d(v) X (t 1). v Y s s Partitio the elemets of A B ito matchigs M 1,..., M, ad let H i be the subhypergraph of H iduced by those edges that cotai some pair of M i. By the pigeohole priciple, H i has more tha c s,t 1/s edges for some i. Let G i be the graph o vertex 15

14 set B C, with edge set {(b, c) : a, (a, b, c) E(H i )}. The by the choice of c s,t, we coclude that G i cotais a copy of K s,t with t vertices i B ad s vertices i C, which exteds via M i to a K () s,t i H. This cotradictio proves z(, K () s,t ) < c s,t 1/s. 6 Upper Boud for ex(, K (),t ) We will use the techiques of Sectio 4 to prove Theorem 1.5. Because our bouds should be thought of as asymptotic results, we omit ceilig ad floor symbols i this sectio. As i Sectio 4, we use Lemma 4.1 to obtai Theorem 1.5 from the followig theorem: Theorem 6.1 For t, z(, K (),t ) < t 5 1/5. Proof. Let H be a -partite triple system, each part of which is a copy of [], labeled A, B, C. As i the proof of Theorem 4., defie the graphs G i = G i (A, B), G = G(A, B), G i = G i(a, C) ad G = G (A, C). Partitio A, B, ad C ito m = /5 disjoit sets A 1, A,..., A m, B 1, B,..., B m, ad C 1,..., C m respectively, so that all sets have size /m = /5. We defie D(A i ) to be the -graph (with multiple edges) o A i such that S A i is a edge of D(A i ) wheever S is a edge of D,t (G j ) = D(G j ), for each j []. We defie D(B i ) o vertex set B i similarly. The first claim is similar to Claim 1 i the proof of Theorem 4.: Claim 1. Ay member of A () B () forms a part of a copy of a K,t i G i for at most t 1 itegers i []. By usig Lemma A.4 i the appedix, we prove the followig Claim i a similar way to Claim i Theorem 4.. The case t = also follows i this way. Claim. For t ad i [m], D(A i ) ad D(B i ) have edge-multiplicity at most 1 ()4. Claim. We may remove at most t 5 m ( ) /m edges from G G so that, for all i [] ad j, k, l [m], the resultig graph cotais o copy of K s,t i G i (A j, B k ) or G i(a j, C l ). 16

15 Proof. Fix j, k [m] ad let H i = G i (A j, B k ). By the Mai Lemma, we may remove at most te(d,t (H i )) edges from H i to obtai a K,t -free graph. As D(A j ) ad D(B k ) have edge-multiplicity at most 1 (t 1)4, the umber of edges which may be removed to delete all copies of K,t i the graphs H i is at most i [] te(d,t (H i )) t 1 ( ) /m (t 1)4 ( ) /m < t 5. Repeatig this argumet for all pairs j, k [m], the umber of edges removed from G is at most t 5 m ( ) /m. Applyig a similar argumet to G gives the same result for G. The total umber of edges removed from G G is therefore at most t 5 m ( ) /m, completig the proof of Claim. We let H G ad H G deote the subgraphs of G ad G remaiig after deletig the edges from G ad G i the applicatio of Claim. Claim 4. e(h G ) < t / 7/ m /. Proof. It is sufficiet to prove that e jk = e(h G (A j, B k )) t / 7/ m 4/ for all j, k [m]. Suppose, for a cotradictio, that e jk > t / 7/ m 4/. The average codegree of pairs (b, c) i B k C is the at least 1 B k C t/ 7/ m 4/ t/ 1/ m 1/. Cosequetly, the iequalities a b /b b ( ) a b < (a) b /b b, ad covexity of biomial coefficiets yield, (b,c) B k C ( ) codeg(b, c) m ( t / 1/ m 1/ ) > t m = t m m > t m ( ) /m. This implies the existece of a set P B k C of t m pairs ad vertices a 1, a, a A j such that the triples (a i, b, c) are edges of H wheever (b, c) P ad i []. Let G deote the bipartite graph o B k C whose edges are the elemets of P. By Lemma A., G cotais a matchig M with t edges or a star with tm edges. I the former case, the set of all triples (a i, b, c) with (b, c) M ad i [] form a copy of K (),t i G. I the latter case, depedig o where the ceter of the star lies, we obtai either (i) a copy of K,mt i G i (A j, B k ) etirely cotaied H G, or 17

16 (ii) a copy of K,mt i G i(a j, C) etirely cotaied i H G. I case (ii), the pigeohole priciple implies that for some l [m], there is (ii ) a copy of K,t i G i(a j, C l ) etirely cotaied i H G. Both (i) ad (ii ) give cotradictios, as all such subgraphs were removed from G G i the applicatio of Claim. Therefore e jk < t / 7/ m 4/, as required. This completes the proof of Claim 4. We ow cout the umber of edges i H. Recallig that m = /5, this umber is at most ( ) /m e(h G ) + t 5 m < t / 7/ m / + t 5 m 6m < t/ 9/ t5 /5 < t 5 1/5. This completes the proof of Theorem 1.5. Remarks: Sice we believe that the expoet 1/5 i Theorem 1.5 ca be improved to 7/, we have made o attempt to optimize the costats i the proof above. This approach gives the upper boud ex(, K () s,t ) < c s,t /[(s 1) +1] for all s, but the boud ex(, K () s,t ) < c s,t 1/s i Theorem 1.4 is better for s >. Appedix The followig two results are due to Füredi (Lemma.1 i [4]): Propositio A.1 Let B be a hypergraph, possibly with multiple edges, i which deg B (v) > Γ B (v) for every vertex v. The e(b) (B). Proof. We prove this by iductio o (B). Let v be ay vertex of B, ad let V be the vertex set of B. Let B be the hypergraph with vertex set V = V Γ B (v) {v} ad edge set {B V : B E(B), B V }. The, by hypothesis ad iductio applied to B, e(b) deg B (v) + e(b ) > deg B (v) + (B ) = (B) 1. This completes the proof. Lemma A. Let s, ad let B 1, B,..., B k be sets of size at least s. The i [k] B (s) i 1 s i [k] B i. 18

17 Proof. Form a hypergraph B with vertex set i [k] B i ad edge set i [k] B (s) i. The, as B i s for i [k], every vertex i B has degree at least oe, so we have (B) = i [k] B i v B deg B (v) = s e(b). Lemma A. Let G be a simple bipartite graph with at least (m 1)(s 1) + 1 edges. The G cotais a matchig with m edges or a star with s edges. Proof. If every vertex of G has degree less tha s, the we require at least m vertices to cover all the edges of G. Hece, by the Köig-Egerváry Theorem (see [1], page 11), G has a matchig with at least m edges. Lemma A.4 Let t > s, µ s, ad let G be a s-graph with e(g) µ ( ) s + 1 ad edge-multiplicity at most µ. The G cotais t vertices, each icidet with a differet edge of G. If s = t, the the same coclusio holds as log as e(g) s. Proof. The case s = t is trivial, so we focus o t > s. Fixig µ s, we will prove the lemma by iductio o t > s. We may assume e(g) = µ ( ) s + 1 ad G cotais o isolated vertices. Let G be the bipartite graph whose parts are the vertex set A of G ad the edge set B of G ad a vertex of G is joied to all the edges of G cotaiig it. Thus every vertex i B has degree s. Therefore Γ G (X) X for all X B with X s. Suppose that t = s + 1. If Γ G (X) s + 1 for some X B with X = s + 1, the we ca apply Hall s Theorem to the bipartite graph iduced by X Γ G (X). This gives s + 1 = t elemets i B matched to t elemets i A. However, such a X exists sice e(g) µ + 1, implies that Γ(B) s + 1, ad this yields a set X B with X = ad Γ(X ) s + 1. Now X ca be exteded to a set X as required. We may therefore assume that t s +. If some vertex v of G has degree at most µ ( ) t s 1, the we remove v from G to obtai a graph G with at least µ ( ) t s + 1 edges. By iductio, there exists a set E of t 1 edges i G ad a set S of t 1 vertices, each icidet with a differet edge i E. As v is ot a isolated vertex i G, we may select ay edge icidet with v, distict from ay edge i E. The S {v} is the required set of t vertices of G. We therefore suppose G cotais o vertex of degree at most µ ( ) t s 1. This implies that for ay X A, s Γ(X) µ ( ) t s 1 X s X. By Hall s Theorem, we may fid a matchig of A ito B. As e(g) > µ ( ) s ad G has edge-multiplicity at most 19

18 µ, G has at least t vertices. Therefore the vertex set of G satisfies the requiremets of the lemma whe t s +. This lemma is best possible as show by the complete s-graph o vertices with every edge repeated µ times. 7 Ackowledgmets We are very thakful to Z. Füredi for helpful commets ad to the referee for carefully readig ad correctig errors i a earlier versio of this mauscript. We would like to thak Microsoft research, where this work was doe while both authors were postdocs. Refereces [1] Alo, N., Róyai, L., Szabó, T., Norm-graphs: variatios ad applicatios. J. Combi. Theory Ser. B 76 (1999), o., [] Bollobás, B., Moder Graph Theory, page 114 [] Erdős, P., Problems ad results i combiatorial aalysis. Proceedigs of the Eighth Southeaster Coferece o Combiatorics, Graph Theory ad Computig (Louisiaa State Uiv., Bato Rouge, La., 1977), 1. Cogressus Numeratium XIX, Utilitas Math., Wiipeg, Ma., [4] Füredi, Z., Hypergraphs i which all disjoit pairs have distict uios. Combiatorica 4 (1984), o. -, [5] Erdős, P., Kleitma, D., O colorig graphs to maximize the proportio of multicolored k-edges. J. Combi. Theory 5 (1968) [6] Füredi, Z., Turá type problems. Surveys i combiatorics, 1991 (Guildford, 1991), 5 00, Lodo Math. Soc. Lecture Note Ser. 166 Cambridge Uiv. Press, Cambridge, [7] Füredi, Z., New asymptotics for bipartite Turá umbers. J. Combi. Theory Ser. A 75 (1996), o. 1,

19 [8] Frakl, P., Extremal set systems. Hadbook of combiatorics, Vol. 1,, 19 19, Elsevier, Amsterdam, [9] Kollár, J., Róyai, L., ad Szabó, T., Norm-graphs ad bipartite Turá umbers. Combiatorica 16 (1996), o., [10] Kővári, T., Sós, V. T., ad Turá, P., O a problem of K. Zarakiewicz, Colloq. Math., (1954), [11] Mubayi, D., Some exact results ad ew asymptotics for hypergraph Turá umbers, accepted i: Combi. Probab. Comput. [1] Ray-Chaudhuri, D., Wilso, R., The existece of resolvable block desigs. Survey of combiatorial theory (Proc. Iterat. Sympos., Colorado State Uiv., Fort Collis, Colo., 1971), pp North-Hollad, Amsterdam, 197. [1] West, D., Itroductio to Graph Theory, d Ed. Pretice-Hall (001). 1