aa r X i v : . [ m a t h . C O ] F e b The number of triple systems without even cycles
Dhruv Mubayi ∗ Lujia Wang † November 10, 2018
Abstract
For k >
4, a loose k -cycle C k is a hypergraph with distinct edges e , e , . . . , e k suchthat consecutive edges (modulo k ) intersect in exactly one vertex and all other pairs ofedges are disjoint. Our main result is that for every even integer k >
4, there exists c > n ] containing no C k is atmost 2 cn .An easy construction shows that the exponent is sharp in order of magnitude. Thismay be viewed as a hypergraph extension of the work of Morris and Saxton, whoproved the analogous result for graphs which was a longstanding problem. For r -uniformhypergraphs with r >
3, we improve the trivial upper bound but fall short of obtainingthe order of magnitude in the exponent, which we conjecture is n r − .Our proof method is different than that used for most recent results of a similar flavorabout enumerating discrete structures, since it does not use hypergraph containers.One novel ingredient is the use of some (new) quantitative estimates for an asymmetricversion of the bipartite canonical Ramsey theorem. An important theme in combinatorics is the enumeration of discrete structures that havecertain properties. Within extremal combinatorics, one of the first influential results ofthis type is the Erd˝os-Kleitman-Rothschild theorem [25], which implies that the number oftriangle-free graphs with vertex set [ n ] is 2 n / o ( n ) . This has resulted in a great deal ofwork on problems about counting the number of graphs with other forbidden subgraphs [6,7, 8, 14, 15, 26, 31, 40, 48] as well as similar question for other discrete structures [10, 11, 17,18, 35, 46, 47, 49, 51]. In extremal graph theory, these results show that such problems areclosely related to the corresponding extremal problems. More precisely, the Tur´an problemasks for the maximum number of edges in a (hyper)graph that does not contain a specificsubgraph. For a given r -uniform hypergraph (henceforth r -graph) F , let ex r ( n, F ) be themaximum number of edges among all r -graphs G on n vertices that contain no copy of ∗ Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607,
Email:[email protected] . Research partially supported by NSF grant DMS-1300138. † Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607,
Email:[email protected] . Research partially supported by NSF grant DMS-1300138. as a (not necessarily induced) subgraph. Henceforth we will call G an F -free r -graph.Write Forb r ( n, F ) for the set of F -free r -graphs with vertex set [ n ]. Since each subgraphof an F -free r -graph is also F -free, we trivially obtain | Forb r ( n, F ) | > ex r ( n,F ) by takingsubgraphs of an F -free r -graph on [ n ] with the maximum number of edges. On the otherhand for fixed r and F , | Forb r ( n, F ) | X i ex r ( n,F ) (cid:18)(cid:0) nr (cid:1) i (cid:19) = 2 O (ex r ( n,F ) log n ) , so the issue at hand is the factor log n in the exponent. The work of Erd˝os-Kleitman-Rothschild [25] and Erd˝os-Frankl-R¨odl [26] for graphs, and Nagle-R¨odl-Schacht [45] forhypergraphs (see also [44] for the case r = 3) improves the upper bound above to obtain | Forb r ( n, F ) | = 2 ex r ( n,F )+ o ( n r ) . Although much work has been done to improve the exponent above (see [1, 6, 7, 8, 31, 34, 48]for graphs and [10, 11, 21, 47, 13, 50] for hypergraphs), this is a somewhat satisfactory stateof affairs when ex r ( n, F ) = Ω( n r ) or F is not r -partite.In the case of r -partite r -graphs, the corresponding questions appear to be more challengingsince the tools used to address the case ex r ( n, F ) = Ω( n r ) like the regularity lemma are notapplicable. The major open problem here when r = 2 is to prove that | Forb r ( n, F ) | = 2 O (ex r ( n,F )) . The two cases that have received the most attention are for r = 2 (graphs) and F = C l or F = K s,t . Classical results of Bondy-Simonovits [16] and Kov´ari-S´os-Tur´an [36] yieldex ( n, C l ) = O ( n /l ) and ex ( n, K s,t ) = O ( n − /s ) for 2 s t , respectively. Althoughit is widely believed that these upper bounds give the correct order of magnitude, this isnot known in all cases. Hence the enumerative results sought in these two cases were | Forb ( n, C l ) | = 2 O ( n /l ) and | Forb ( n, K s,t ) | = 2 O ( n − /s ) . In 1982, Kleitman and Winston [32] proved that | Forb ( n, C ) | = 2 O ( n / ) which initi-ated a 30-year effort on searching for generalizations of the result to complete bipar-tite graphs and even cycles. Kleitman and Wilson [33] proved similar results for C and C in 1996 by reducing to the C case. Finally, Morris and Saxton [42] recentlyproved that | Forb ( n, C l ) | = 2 O ( n /l ) and Balogh and Samotij [14, 15] proved that | Forb ( n, K s,t ) | = 2 O ( n − /s ) for 2 s t . Both these results used the hypergraph containermethod (developed independently by Saxton and Thomason [50], and by Balogh-Morris-Samotij [13]) which has proved to be a very powerful technique in extremal combinatorics.For example, [13] and [50] reproved | Forb r ( n, F ) | = 2 ex r ( n,F )+ o ( n r ) using containers.There are very few results in this area when r > r ( n, F ) = o ( n r ). The only casessolved so far are when F consists of just two edges that intersect in at least t vertices [9], orwhen F consists of three edges such that the union of the first two is equal to the third [12](see also [4, 5, 22, 23] for some related results). These are natural points to begin theseinvestigations since the corresponding extremal problems have been studied deeply.2ecently, Kostochka, the first author and Verstra¨ete [37, 38, 39], and independently, F¨urediand Jiang [29] (see also [30]) determined the Tur´an number for several other families of r -graphs including paths, cycles, trees, and expansions of graphs. These hypergraph ex-tremal problems have proved to be quite difficult, and include some longstanding conjec-tures. Guided and motivated by these recent developments on the extremal number ofhypergraphs, we consider the corresponding enumeration problems focusing on the case ofcycles. Definition 1
For each integer k > , a k -cycle C k is a hypergraph with distinct edges e , . . . , e k and distinct vertices v , . . . , v k such that e i ∩ e i +1 = { v i } for all i k − , e ∩ e k = { v k } and e i ∩ e j = ∅ for all other pairs i, j . Sometimes we refer to C k as a loose or linear cycle. To simplify notation, we will omit theparameter r when the cycle C k is a subgraph of an r -graph.Since ex r ( n, C k ) = O ( n r − ), we obtain the upper bound | Forb r ( n, C k ) | = 2 O ( n r − log n ) when r and k are fixed and n → ∞ . Our main result is the following theorem, whichimproves this upper bound and generalizes the Morris-Saxton theorem [42] to 3-graphs. Theorem 2 (Main Result)
For integers r, k > , there exists c = c ( r, k ) , such that | Forb r ( n, C k ) | < ( c n if r = 3 and k is even, c n r − (log n ) ( r − / ( r − if r > . Since trivially ex r ( n, C k ) = Ω( n r − ) for all r >
3, we obtain | Forb ( n, C k ) | = 2 Θ( n ) when k is even. We conjecture that a similar result holds for r > Conjecture 3
For fixed r > and k > we have | Forb r ( n, C k ) | = 2 Θ( n r − ) . Almost all recent developments in this area have relied on the method of hypergraph con-tainers that we mentioned above. What is perhaps surprising about the current work isthat the proofs do not use hypergraph containers. Instead, our methods employ old andnew tools in extremal (hyper)graph theory. The old tools include the extremal numbersfor cycles modulo h and results about decomposing complete r -graphs into r -partite ones,and the new tools include the analysis of the shadow for extremal hypergraph problems andquantitative estimates for the bipartite canonical Ramsey problem. Throughout this paper, we let [ n ] denote the set { , , . . . , n } . Write (cid:0) Xr (cid:1) = { S ⊂ X : | S | = r } and (cid:0) X r (cid:1) = { S ⊂ X : | S | r } . For X ⊂ [ n ], an r -uniform hypergraph or r -graph H on vertex set X is a collection of r -element subsets of X , i.e. H ⊂ (cid:0) Xr (cid:1) . The vertex3et X is denoted by V ( H ). The r -sets contained in H are edges . The size of H is | H | .Given S ⊂ V ( H ), the neighborhood N H ( S ) of S is the set of all T ⊂ V ( H ) \ S such that S ∪ T ∈ H . The codegree of S is d H ( S ) = | N H ( S ) | . When the underlying hypergraph isclear from context, we may omit the subscripts in these definitions and write N ( S ) and d ( S ) for simplicity. The sub-edges of H are the ( r − n ] with positive codegreein H . The set of all sub-edges of H is called the shadow of H , and is denoted ∂H .An r -partite r -graph H is an r -graph with vertex set F ri =1 V i (the V i s are pairwise disjoint),and every e ∈ H satisfies | e ∩ V i | = 1 for all i ∈ [ r ]. When all such edges e are present, H is called a complete r -partite r -graph . When | V i | = s for all i ∈ [ r ], a complete r -partite r -graph H is said to be balanced , and denoted K s : r .For each integer k >
1, a (loose, or linear) path of length k denoted by P k , is a collection of k edges e , e , . . . , e k such that | e i ∩ e j | = 1 if i = j + 1, and e i ∩ e j = ∅ otherwise.We will often omit floors and ceilings in our calculations for ease of notation and all logswill have base 2. The proof of Theorem 2 proceeds by counting edge-colored ( r − r = 3 and r >
3. In thissection we state the main technical statement (Theorem 5) about these edge-colorings thatwill be needed, as well as some other tools, and then prove Theorem 2 using these results.
Given an ( r − G with V ( G ) ⊂ [ n ], a coloring function is a function χ : G → [ n ]such that χ ( e ) = z e ∈ [ n ] \ e for every e ∈ G . We call z e the color of e . The vector ofcolors N G = ( z e ) e ∈ G is called an edge-coloring of G . The pair ( G, N G ) is an edge-colored ( r − color class is the set of all edges that receive the same color.Given G , each edge-coloring N G defines an r -graph H ( N G ) = { e ∪ { z e } : e ∈ G } , calledthe extension of G by N G . When there is only one coloring that has been defined, we alsouse the notation G ∗ = H ( N G ) for the extension. Observe that any subgraph G ′ ⊂ G alsoadmits an extension by N G , namely, G ′∗ = { e ∪ { z e } : e ∈ G ′ } ⊂ G ∗ . If G ′ ⊂ G and χ | G ′ is one-to-one, then G ′ is called rainbow colored . If a rainbow colored G ′ further satisfiesthat z e / ∈ V ( G ′ ) for all e ∈ G ′ , then G ′ is said to be strongly rainbow colored . Note that astrongly rainbow colored graph C k ⊂ G ′ gives rise to 3-graph C k in G ′∗ ⊂ G ∗ . Definition 4
Given r > , k > , s > , let f r ( n, k, s ) be the number of edge-coloredbalanced complete ( r − -partite ( r − -graphs G = K s : r − with V ( G ) ⊂ [ n ] , whose extension G ∗ is C k -free. The function f r ( n, k, s ) allows us to encode r -graphs, and our main technical theorem givesan upper bound for this function. 4 heorem 5 Given r > , k > there exist D = D k/ , c = c ( r, k ) , such that f r ( n, k, s ) ( (5 / ks log n +4 s log D if r = 3 , k is even , ( c +2 r ) s r − log n + s r − (log( c + r )+( r −
2) log s ) if r > . For r and k fixed, the bounds above can be written as f r ( n, k, s ) = ( O ( s log n + s ) if r = 3 , k is even,2 O ( s r − log n + s r − log s ) if r > . Note that the trivial upper bound is f r ( n, k, s ) n ( r − s + s r − ∼ s r − log n (first choose( r − s vertices, then color each of its s r − edges) so Theorem 5 is nontrivial only if s = o ( n ). The proof of Theorem 5 will be given in Sections 3–6. r -graphs into balanced complete r -partite r -graphs Chung-Erd˝os-Spencer [19] and Bublitz [3] proved that the complete graph K n can be de-composed into balanced complete bipartite graphs such that the sum of the sizes of thevertex sets in these bipartite graphs is at most O ( n / log n ). See also [55, 43] for somegeneralizations and algorithmic consequences. We need the following generalization of thisresult to r -graphs. Theorem 6
Let n > r > . There exists a constant c ′ = c ′ ( r ) , such that any n -vertex r -graph H can be decomposed into balanced complete r -partite r -graphs K s i : r , i = 1 , . . . , m ,with s i (log n ) / ( r − and P mi =1 s r − i c ′ n r / (log n ) / ( r − .Proof. An old result of Erd˝os [27] states that for any integers r, s > n > rs , we haveex r ( n, K s : r ) < n r − /s r − . Note that for r = 2, this was proved much earlier by K˝ov´ari-S´os-Tur´an [36].We first assume that n > r . Taking the derivative, one can show that for each r > n n/r − (log n ) / ( r − is an increasing function in n , hence its minimum is achieved at n = 2 r . So for all n > r , we have nr − (log n ) / ( r − > rr − (log 2 r ) / ( r − = 2 − (log 2 r ) / ( r − > . Thus, for any s (log n ) / ( r − n/r , the Tur´an number for K s : r isex r ( n, K s : r ) < n r − /s r − . Next, we give an algorithm of decomposing H into K s : r s. Let H = H . For i >
1, repeatedlyfind a K s i : r ⊂ H i with maximum s i subject to s i (log n ) / ( r − and delete it from H i toform H i +1 . The loop terminates at step i if | H i | n r / (log n ) / ( r − . Then let the remaininggraph be decomposed into single edges ( K r s).5y the algorithm, the vertex size of each K s i : r satisfies that s i (log n ) / ( r − , i m automatically. So we are left to show the upper bound for P mi =1 s r − i .We divide the iterations of the above algorithm into phases, where the k th phase consistsof those iterations where the number of edges in the input r -graph of the algorithm lies inthe interval ( n r / ( k + 1) , n r /k ]. In other words, in phase k , each K s i : r to be found is in an r -graph H i with | H i | > n r / ( k + 1). Define s ( k ) = (log n/ log( k + 1)) / ( r − . Then it is easyto see s ( k ) (log n ) / ( r − n/r . So by Erd˝os’ result n r k + 1 = n r − /s ( k ) r − > ex( n, K s ( k ): r ) . Hence, K s ( k ): r ⊂ H i . So in phase k , the minimum s i of a K s i : r we are able to find has thelower bound s i > s ( k ) = (cid:18) log n log( k + 1) (cid:19) / ( r − . Now notice that m X i =1 s r − i = m X i =1 s ri · s i . Dividing up the terms in the summation according to phases, we observe that this is a sumof the number of edges deleted in the k th phase times a weight of 1 /s i for each edge. Alsonotice that there are at most n r / (log n ) / ( r − single edges, we have m X i =1 s r − i n r (log n ) / ( r − + ∞ X k =1 n r (cid:18) k − k + 1 (cid:19) (cid:18) log( k + 1)log n (cid:19) / ( r − = n r (log n ) / ( r − + ∞ X k =1 n r (log( k + 1)) / ( r − k ( k + 1)(log n ) / ( r − = c ′ n r (log n ) / ( r − , where c ′ = 1 + ∞ X k =1 (log( k + 1)) / ( r − k ( k + 1) . Finally, for n < r , we just let H be decomposed into K r s. Then we obtain the followingbound for P mi =1 s r − i . m X i =1 s r − i = m (cid:18) rr (cid:19) c ′ n r (log n ) / ( r − with appropriately chosen c ′ = c ′ ( r ). This completes the proof.6 .3 A corollary of Theorem 5 Theorem 5 is about the number of ways to edge-color complete ( r − r − s and vertex set in [ n ]. In this section, we use Theorems 5 and 6 to provea related statement where we do not require the ( r − s vertices. Definition 7
For r > and k > , let g r ( n, k ) be the number of edge-colored ( r − -graphs G with V ( G ) ⊂ [ n ] such that the extension G ∗ is C k -free. Lemma 8
Let r > , k > , and n be large enough. Then there exist c = c ( r ) , c = c ( r, k ) and D = D k/ , such that g r ( n, k ) ( (3 kc +4 log D ) n if r = 3 , k is even, c +2 r ) c n r − (log n ) ( r − / ( r − if r > . Note that if r and k are fixed, for both cases we have g r ( n, k ) = 2 O ( n r − (log n ) ( r − / ( r − ) . Proof.
Given any ( r − G , by applying Theorem 6 with parameter r − r , we may decompose G into balanced complete ( r − r − K s : r − , . . . , K s m : r − , with s i (log n ) / ( r − and P mi =1 s r − i c n r − / (log n ) / ( r − , where c = c ( r ) = c ′ ( r − • From the second inequality, we have m c n r − / (log n ) / ( r − . • Using the fact that these copies of K s i : r − are edge disjoint, we have m X i =1 s r − i (cid:18) nr − (cid:19) < n r − . Therefore, to construct an edge-colored G , we need to first choose a sequence of positiveintegers ( m, s , . . . , s m ) such that m c n r − / (log n ) / ( r − , and s i (log n ) / ( r − for all i . More formally, let S n,r = { ( m, s , s , . . . , s m ) : m c n r − / (log n ) / ( r − , s i (log n ) / ( r − , i m } . Then | S n,r | c n r − (log n ) / ( r − (cid:16) (log n ) / ( r − (cid:17) c nr − n )1 / ( r − = 2 log (cid:18) c nr − n )1 / ( r − (cid:19) + c nr − n )1 / ( r − n )1 / ( r − c n r − . K s i : r − for each i ∈ [ m ]. To make sure G ∗ is C k -free, K ∗ s i : r − has to be made C k -free in the first place. Applying Theorem 5, we get thefollowing upper bounds. For r = 3 and even k , g ( n, k ) X ( m,s ,...,s m ) ∈ S n, m Y i =1 f ( n, k, s i ) X ( m,s ,...,s m ) ∈ S n, m Y i =1 (5 / ks i log n +4 s i log D X ( m,s ,...,s m ) ∈ S n, P mi =1 (5 / ks i log n +4 s i log D X ( m,s ,...,s m ) ∈ S n, (5 / k log n ( c n / log n )+4 n log D c n · (5 / kc n +4 n log D (3 kc +4 log D ) n . For r >
3, and ( m, s , . . . , s m ) ∈ S n,r , the number of ways to construct these copies of K s i : r − is at most m Y i =1 f r ( n, k, s i ) m Y i =1 ( c +2 r ) s r − i log n + s r − i (log( c + r − r −
2) log s i ) = 2 P mi =1 ( c +2 r ) s r − i log n + s r − i (log( c + r − r −
2) log s i ) ( c +2 r ) c nr − n )1 / ( r − log n + n r − (log( c + r − r −
2) log(log n ) / ( r − ) ( c +2 r ) c n r − (log n ) ( r − / ( r − + n r − (log( c + r − r −
2) log(log n ) / ( r − ) (3 / c +2 r ) c n r − (log n ) ( r − / ( r − . Note that this is the only place in the proof where we use s i (log n ) / ( r − .Therefore, g r ( n, k ) X ( m,s ,...,s m ) ∈ S n,r m Y i =1 f r ( n, k, s i ) c n r − · (3 / c +2 r ) c n r − (log n ) ( r − / ( r − c +2 r ) c n r − (log n ) ( r − / ( r − , and the proof is complete. A crucial statement that we use in our proof is that any r -graph such that every sub-edgehas high codegree contains rich structures, including cycles. This was explicitly provedin [37] and we reproduce the proof here for completeness.8 emma 9 (Lemma 3.2 in [37]) For r, k > , if all sub-edges of an r -graph H havecodegree greater than rk , then C k ⊂ H .Proof. Let F = ∂ r − H be the (2-)graph that consists of pairs that are contained in someedge of H . Note that each edge of H induces a K r in F , so all edges of F are contained insome triangle ( C ). Furthermore, since all sub-edges of H have codegree greater than rk ,each edge of F is in more than rk triangles. We will first find a k -cycle in F as follows.Starting with a triangle C , for i = 3 , . . . , k − e ∈ C i , form C i +1 by replacing e by the other two edges of one of at least rk − i + 2 triangles containing e and excludingother vertices of C i .Next, let C k ⊂ F be a k -cycle with edges f , . . . , f k . Find in H a subgraph C = { e i : e i = f i ∪ g i , i ∈ [ k ] } such that V ( C ) = ∪ ki =1 e i is of maximum size. Suppose C is not a k -cyclein H . Then there are distinct i, j such that g i ∩ g j = ∅ . Pick v ∈ g i ∩ g j and consider thesub-edge e i \{ v } = f i ∪ g i \{ v } . The codegree d H ( e i \{ v } ) > rk by assumption. On the otherhand, | V ( C ) | < rk since C is not a k -cycle, so there exists a vertex v ′ ∈ N H ( e i \ { v } ) \ V ( C ).Replacing e i by e i \ { v } ∪ { v ′ } , we obtain a C ′ with a larger vertex set, a contradiction. So H contains a C k . Now we have all the ingredients to complete the proof of our main result.
Proof of Theorem 2.
Starting with any r -graph H on [ n ] with C k H , we claim thatthere exists a sub-edge with codegree at most rk . Indeed, otherwise all sub-edges of H willhave codegree more than rk , and then by Lemma 9 we obtain a C k ⊂ H . Let e ′ be thesub-edge of H with 0 < d H ( e ′ ) rk such that it has smallest lexicographic order among allsuch sub-edges. Delete all edges of H containing e ′ from H (i.e. delete { e ∈ H : e ′ ⊂ e } ).Repeat this process of “searching and deleting” in the remaining r -graph until there are nosuch sub-edges. We claim that the remaining r -graph must have no edges at all. Indeed,otherwise we get a nonempty subgraph all of whose sub-edges have codegree greater than rk , and again by Lemma 9, we obtain a C k ⊂ H .Given any C k -free r -graph H on [ n ], the algorithm above sequentially decomposes H into acollection of sets of at most rk edges who share a sub-edge (an ( r − r − r − G . Moreover, for each edge e ∈ G , let N e be the set of vertices v ∈ V ( H ) such that e ∪ { v } is an edge of H at the time e was chosen. So N e ∈ (cid:0) [ n ] \ e rk (cid:1) , for all e ∈ G . Thus, we get a map φ : Forb r ( n, C k ) −→ (cid:26) ( G, N G ) : G ⊂ (cid:18) [ n ] r − (cid:19) , N G = (cid:18) N e ∈ (cid:18) [ n ] \ e rk (cid:19) : e ∈ G (cid:19)(cid:27) . We observe that φ is injective. Indeed, φ − (( G, N G )) = H ( N G ) = { e ∪ { z e } : e ∈ G, z e ∈ N e } , | Forb r ( n, C k ) | = | φ (Forb r ( n, C k )) | . Let P = φ (Forb r ( n, C k )) which is the set ofall pairs ( G, N G ) such that H ( N G ) is C k -free. Next we describe our strategy for upperbounding | P | .For each pair ( G, N G ) ∈ P and e ∈ G , we pick exactly one z e ∈ N e . Thus we get a pair( G , N G ), where G = G , and N G = ( z e : e ∈ G ). Then, delete z e from each N e , let G = { e ∈ G : N e \ { z e } 6 = ∅} and pick z e ∈ N e \ { z e } to get the pair ( G , N G ). For2 i < rk , we repeat this process for G i to obtain G i +1 . Since each N G i contains onlysingletons, the pair ( G i , N G i ) can be regarded as an edge-colored ( r − G i s. This gives us a map ψ : P −→ (cid:26) ( G , . . . , G rk ) : G i ⊂ (cid:18) [ n ] r − (cid:19) is edge-colored for all i ∈ [ rk ] (cid:27) . Moreover, it is almost trivial to observe that ψ is injective, since if y = y ′ , then either theunderlying ( r − y and y ′ differ, or the ( r − ψ ( y ) = ψ ( y ′ ). Again, we let Q = ψ ( P ).By Lemma 8, we have | Forb r ( n, C k ) | = | P | = | Q | rk Y i =1 g r ( n, k ) (Q ki =1 (3 kc +4 log D ) n if r = 3 , k is even Q rki =1 c +2 r ) c n r − (log n ) ( r − / ( r − if r > ( P ki =1 (3 kc +4 log D ) n if r = 3 , k is even2 P rki =1 c +2 r ) c n r − (log n ) ( r − / ( r − if r > ( (9 k c +12 k log D ) n if r = 3 , k is even2 rk ( c +2 r ) c n r − (log n ) ( r − / ( r − if r >
3= 2 c n r − (log n ) ( r − / ( r − . where c = 9 k c + 12 k log D if r = 3 and k is even, and c = 2 rk ( c + 2 r ) c if r >
3. Theconstants c = c ( r ) , c = c ( r, k ) and D = D k/ are from Theorem 6 and Theorem 5,respectively. r > In the remaining part of the paper, we prove Theorem 5. The cases r = 3 and r > r > C l . This is a corollary of the main result of [37] although it can also be proveddirectly by analyzing the shadow with a much better bound. Lemma 10
Let r > , k > and G be an r -partite r -graph on vertex sets F ri =1 V i with | V i | = s, for all i . There exists c ′ = c ′ ( r, k ) , such that if | G | > c ′ s r − then G contains acycle of length k . roof. By Theorem 1.1 of [37], we haveex r ( n, C k ) ∼ (cid:18) nr (cid:19) − (cid:18) n − ⌊ ( k − / ⌋ r (cid:19) ∼ ⌊ ( k − / ⌋ ( r − n r − . So, we may take c ′ large enough such that c ′ s r − = c ′ r r − n r − > ⌊ ( k − / ⌋ ( r − n r − , to guarantee the existence of a copy of C k in G . We remark that c ′ = ⌊ k/ ⌋ r r − ( r − suffices. Proof of Theorem 5 for r > . Let G = K s : r − with V ( G ) ⊂ [ n ] and C k G ∗ . Forany edge-coloring N G = ( z e : e ∈ G ) of G , let Z = { z e : e ∈ G } ⊂ [ n ] be the set of allits colors. We first argue that | Z | < ( c + r − s r − , where c = c ( r, k ) = c ′ ( r − , k ) = ⌊ k/ ⌋ ( r − r − / ( r − | Z | > ( c + r − s r − ,then | Z \ V ( G ) | > ( c + r − s r − − s ( r − > ( c + r − − ( r − s r − = c s r − . Foreach color v ∈ Z \ V ( G ) pick an edge in G with color v . We get a subgraph G ′ ⊂ G that isstrongly rainbow with | G ′ | = | Z \ V ( G ) | > c s r − . By Lemma 10, we find an ( r − C k in G that is strongly rainbow, which contradicts the fact that C k G ∗ .We now count the number of edge-colored K s : r − as follows: first choose s ( r −
1) verticesfrom [ n ] as the vertex set, then choose at most ( c + r − s r − colors, finally color eachedge of the K s : r − . As | K s : r − | = s r − , this yields f r ( n, k, s ) n s ( r − c + r − s r − (( c + r − s r − ) s r − = 2 ( s ( r − c + r − s r − ) log n + s r − (log( c + r − r −
2) log s ) ( c +2 r ) s r − log n + s r − (log( c + r − r −
2) log s ) . r = 3 and even k The rest of the paper is devoted to the proof of Theorem 5 for r = 3 and even k . Forsimplicity of presentation, we write k = 2 l where l >
2. Our two main tools are thefollowing lemmas about edge-coloring bipartite graphs.
Lemma 11
Let l > , s, t > , G = K s,t be an edge-colored complete bipartite graph with V ( G ) ⊂ [ n ] and Z = { z e : e ∈ G } ⊂ [ n ] be the set of all colors. If G contains no stronglyrainbow colored C l , then | Z | < l ( s + t ) . Lemma 12
For each l > , there exists a constant D = D l > , such that the followingholds. Let s, t > , G = K s,t be a complete bipartite graph with vertex set V ( G ) ⊂ [ n ] , and Z ⊂ [ n ] be a set of colors. Then the number of ways to edge-color G with Z such that theextension G ∗ contains no C l , is at most D ( s + t ) . r = 3 and even k . Proof of Theorem 5 for r = 3 and k = 2 l . Recall that r > l >
2, and that f ( n, l, s ) isthe number of edge-colored copies of K s,s whose vertex set lies in [ n ] and whose (3-uniform)extension is C l -free. To obtain such a copy of K s,s , we first choose from [ n ] its 2 s vertices,then its at most 4 ls colors by Lemma 11 and finally we color this K s,s by Lemma 12. Thisyields f ( n, l, s ) n s +4 ls D (2 s ) ls log n +4 s log D = 2 (5 / ks log n +4 s log D , where the last inequality holds since l >
2. Note that D = D l = D k/ is the desiredconstant. In this section we prove Lemma 11. Our main tool is an extremal result about cycles modulo h in a graph. This problem has a long history, beginning with a Conjecture of Burr andErd˝os that was solved by Bollob´as [2] in 1976 via the following result: for each integer m and odd positive integer h , every graph G with minimum degree δ ( G ) > h + 1) h − /h contains a cycle of length congruent to m modulo h . The lower bound on δ ( G ) was improvedby Thomassen [53] who also generalized it to the case with all integers h . It was conjecturedby Thomassen that graphs with minimum degree at least h + 1 contain a cycle of length 2 m modulo h , for any h, m >
1. The conjecture received new attention recently. In particularLiu-Ma [41] settled the case when h is even, and Diwan [24] proved it for m = 2. Todate, Sudakov and Verstra¨ete [52] hold the best known bound for the general case on thisproblem.We need the very special case m = 1 of Thomassen’s conjecture and in order to be selfcontained, we give a proof below. The idea behind this proof can be found in Diwan [24]. Lemma 13 If G is an n -vertex graph with at least ( h + 1) n edges, then G contains a cycleof length modulo h .Proof. By removing vertices of degree at most h , we may assume that G has minimumdegree at least h + 1. Let P be a longest path in G . Assume that P is of length l . Let V ( P ) = { x , x , . . . , x l } , where x , x l are the two end-vertices of P , and x i − x i ∈ P for all i ∈ [ l ]. Then we observe that N ( x ) ⊂ V ( P ). Otherwise we can extend P to a longer pathby x y with some vertex y ∈ N ( x ) \ V ( P ). So N ( x ) = { x i : i ∈ I } for some I ⊂ [ l ]. Notethat 1 ∈ I , | I | = | N ( x ) | > δ ( G ) > h + 1, and the distance dist P ( x i , x j ) = | j − i | . Considerthe set J = { i − i ∈ I, i = 1 } , note that this is the set of all distances dist P ( x , x i ) with x i ∈ N ( x ) and i = 1. Clearly, | J | = | I | − > h . If there exists i − ∈ J with i − ≡ h ), then we are done, since the sub-path of P from x to x i together with x x , x x i form a cycle of length 2 modulo h . So none of the numbers in J are multiples of h . By thepigeonhole principle, there are at least two elements i − , j − ∈ J such that i − ≡ j − h ), thus dist P ( x i , x j ) = | j − i | ≡ h ). Again, we can find a cycle of length 2modulo h by taking the sub-path of P connecting x i , x j and edges x x i , x x j .12 emma 14 Let integers l > , s, t > , G = K s,t with V ( G ) ⊂ [ n ] be edge-colored. If G contains a strongly rainbow colored cycle of length l − , then G contains astrongly rainbow colored C l .Proof. Let us assume that C is the shortest strongly rainbow colored cycle of length 2modulo 2 l − G . Then C has at least 2 l edges. We claim that C is a C l . Suppose not,let e be a chord of C (such a chord exists as G is complete bipartite), such that C is cutup into two paths P and P by the two endpoints of e , and | P | = 2 l −
1. Let Z , Z bethe set of their colors respectively. If the color z e / ∈ Z ∪ V ( P ) \ e , then P ∪ e is a stronglyrainbow colored cycle of length 2 l , a contradiction. Therefore z e ∈ Z ∪ V ( P ) \ e , but then z e / ∈ Z ∪ V ( P ) \ e , yielding a shorter strongly rainbow colored cycle P ∪ e of length 2modulo 2 l −
2, a contradiction.We now have all the necessary ingredients to prove Lemma 11.
Proof of Lemma 11.
Suppose that | Z | > l ( s + t ). Then | Z \ V ( G ) | > (2 l − s + t ). Foreach color v in Z \ V ( G ), pick an edge e of G with color v . We obtain a strongly rainbowcolored subgraph G ′ of G with at least (2 l − s + t ) edges. Lemma 13 guarantees theexistence of a rainbow colored cycle of length 2 modulo 2 l − G ′ . By construction, thiscycle is strongly rainbow. Lemma 14 then implies that there is a strongly rainbow colored C l in G . Our proof of Lemma 12 is inspired by the methods developed in [37]. The main idea isto use the bipartite canonical Ramsey theorem. In order to use this approach we needto develop some new quantitative estimates for an asymmetric version of the bipartitecanonical Ramsey theorem.
In this section we state and prove the main result in Ramsey theory that we will use toprove Lemma 12. We are interested in counting the number of edge-colorings of a bipartitegraph, such that the (3-uniform) extension contains no copy of C l . The canonical Ramseytheorem allows us to find nice colored structures that are easier to work with. However, thequantitative aspects are important for our application and consequently we need to provevarious bounds for bipartite canonical Ramsey numbers. We begin with some definitions.Let G be a bipartite graph on vertex set with bipartition X ⊔ Y . For any subsets X ′ ⊂ X , Y ′ ⊂ Y , let E G ( X ′ , Y ′ ) = G [ X ′ ⊔ Y ′ ] = { xy ∈ G : x ∈ X ′ , y ∈ Y ′ } , and e G ( X ′ , Y ′ ) = | E G ( X ′ , Y ′ ) | . If X ′ contains a single vertex x , then E G ( { x } , Y ′ ) will be simply written as E G ( x, Y ′ ). The subscript G may be omitted if it is obvious from context. Definition 15
Let G be an edge-colored bipartite graph with V ( G ) = X ⊔ Y . G is monochromatic if all edges in E ( X, Y ) are colored by the same color. • G is weakly X -canonical if E ( x, Y ) is monochromatic for each x ∈ X . • G is X -canonical if it is weakly X -canonical and for all distinct x, x ′ ∈ X the colorsused on E ( x, Y ) and E ( x ′ , Y ) are all different.In all these cases, the color z x of the edges in E ( x, Y ) is called a canonical color . Lemma 16
Let G = K a,b be an edge-colored complete bipartite graph with bipartition A ⊔ B ,with | A | = a, | B | = b . If G is weakly A -canonical, then there exists a subset A ′ ⊂ A with | A ′ | = √ a such that G [ A ′ ⊔ B ] = K √ a,b is A ′ -canonical or monochromatic.Proof. Take a maximal subset A ′ of A such that the coloring on E ( A ′ , B ) is A ′ -canonical.If | A ′ | > √ a , then we are done. So, we may assume that | A ′ | < √ a . By maximalityof A ′ , there are less then √ a canonical colors. By the pigeonhole principle, there are atleast | A | / | A ′ | > a/ √ a = √ a vertices of A sharing the same canonical color, which gives amonochromatic K √ a,b .Our next lemma guarantees that in an “almost” rainbow colored complete bipartite graph,there exists a rainbow complete bipartite graph. Lemma 17
For any integer c > , and p > c , if G = K p,p is an edge-colored completebipartite graph, in which each color class is a matching, then G contains a rainbow colored K c,c .Proof. Let A ⊔ B be the vertex set of G . Pick two c -sets X, Y from A and B respectively atrandom with uniform probability. For any pair of monochromatic edges e, e ′ , the probabilitythat they both appear in the induced subgraph E ( X, Y ) is (cid:0) p − c − (cid:1)(cid:0) pc (cid:1) ! = (cid:18) c ( c − p ( p − (cid:19) . On the other hand, the total number of pairs of monochromatic edges is at most p /
2, sinceevery color class is a matching. Therefore the union bound shows that, when p > c , theprobability that there exists a monochromatic pair of edges in E ( X, Y ) is at most p (cid:18) c ( c − p ( p − (cid:19) = pc p − < . Consequently, there exists a choice of X and Y such that the E ( X, Y ) contains no pair ofmonochromatic edges. Such an E ( X, Y ) is a rainbow colored K c,c .Now we are ready to prove the main result of this section which is a quantitative version ofa result from [38]. Note that the edge-coloring in this result uses an arbitrary set of colors.Since the conclusion is about “rainbow” instead of “strongly rainbow”, it is not essential tohave the set of colors disjoint from the vertex set of the graph.14 heorem 18 (Asymmetric bipartite canonical Ramsey theorem) For any integer l > ,there exists real numbers ǫ = ǫ ( l ) > , s = s ( l ) > , such that if G = K s,t is an edge-colored complete bipartite graph on vertex set X ⊔ Y with | X | = s, | Y | = t with s > s and s/ log s < t s , then one of the following holds: • G contains a rainbow colored K l, l , • G contains a K q, l on vertex set Q ⊔ R , with | Q | = q, | R | = 2 l that is Q -canonical,where q = s ǫ , • G contains a monochromatic K q, l on vertex set Q ⊔ R , with | Q | = q, | R | = 2 l , where q = s ǫ .Note that in the last two cases, it could be Q ⊂ X, R ⊂ Y or the other way around.Proof. We will show that ǫ = 1 / l . First, fix a subset Y ′ of Y with | Y ′ | = t / l and let W = (cid:26) x ∈ X : there exists a Y ′′ ∈ (cid:18) Y ′ l (cid:19) such that E G ( x, Y ′′ ) is monochromatic (cid:27) . If | W | > s/ l , then the number of Y ′′ ∈ (cid:0) Y ′ l (cid:1) such that E G ( x, Y ′′ ) is monochromatic forsome x (with repetition) is greater than s/ l . On the other hand, | (cid:0) Y ′ l (cid:1) | < | Y ′ | l = √ t . Bythe pigeonhole principle, there exists a Y ′′ ∈ (cid:0) Y ′ l (cid:1) such that at least s l √ t > s l √ s > s / vertices x have the property that E G ( x, Y ′′ ) is monochromatic. Let Q be a set of s / such x . Then we obtain a weakly Q -canonical K s / , l on Q ⊔ Y ′′ which, by Lemma 16, containsa canonical or monochromatic K s / , l . Since ǫ < /
6, this contains a K s ǫ , l as desired.We may now assume that | W | s/ l . By definition of W and the pigeonhole principle, E G ( x, Y ′ ) contains at least | Y ′ | / l (distinct) colors for every x ∈ X \ W . Hence, for each x ∈ X \ W we can take | Y ′ | / l distinctly colored edges from E ( x, Y ′ ) to obtain a subgraph G ′ of G on ( X \ W ) ⊔ Y ′ with | X \ W || Y ′ | / l edges.Pick a subset X ′ ⊂ X \ W with | X ′ | = s / l and e G ′ ( X ′ , Y ′ ) > | X ′ || Y ′ | / l . This is possibleby an easy averaging argument. Let Z = (cid:26) y ∈ Y ′ : there exists an X ′′ ∈ (cid:18) X ′ l (cid:19) such that E G ′ ( X ′′ , y ) is monochromatic (cid:27) . If | Z | > | Y ′ | / l , then the number of X ′′ ∈ (cid:0) X ′ l (cid:1) such that E G ′ ( X ′′ , y ) is monochromatic forsome y (with repetition) is greater than | Y ′ | / l . On the other hand, | (cid:0) X ′ l (cid:1) | < | X ′ | l = s / l .By the pigeonhole principle, there exists a X ′′ ∈ (cid:0) X ′ l (cid:1) such that at least | Y ′ | ls / l = t / l ls / l > s / l (log s ) / l ls / l > s / l = s ǫ y have the property that E G ′ ( X ′′ , y ) is monochromatic. Let Q be a set of s ǫ such y . We find a weakly Q -canonical K l,s ǫ on X ′′ ⊔ Q . Again, by Lemma 16, a copy of K l,s ǫ that is monochromatic or canonical is obtained.Finally, we may assume that | Z | | Y ′ | / l . Then e G ′ ( X ′ , Y ′ \ Z ) > e G ′ ( X ′ , Y ′ ) − | X ′ || Z | > l | X ′ || Y ′ | − l | X ′ || Y ′ | = 920 l | X ′ || Y ′ | > l | X ′ || Y ′ \ Z | . Since each vertex y ∈ Y ′ \ Z has the property that E G ′ ( X ′ , y ) sees each color at most 2 l − y ∈ Y ′ \ Z we may remove all edges from E G ′ ( X ′ , y ) with duplicated colors(keep one for each color). We end up getting a bipartite graph G ′′ on X ′ ⊔ ( Y ′ \ Z ) with atleast 9 | X ′ || Y ′ \ Z | / l edges. By the K˝ov´ari-S´os-Tur´an theorem [36], there is a c > G ′′ contains a copy K of K p,p where p > c log s . Let V ( K ) = A ⊔ B . For each x ∈ A ,the edges set E ( x, B ) is rainbow colored, and for each y ∈ B , the edge set E ( A, y ) is rainbowcolored. Therefore each color class in K is a matching. By Lemma 17 and s > s > (4 l ) /c ,we can find a rainbow colored K l, l in K as desired. We are now ready to prove Lemma 12. Let us recall the statement.
Lemma 12
For each l > , there exists a constant D = D l > , such that the followingholds. Let s, t > , G = K s,t be a complete bipartite graph with vertex set V ( G ) ⊂ [ n ] , and Z ⊂ [ n ] be a set of colors. Then the number of ways to edge-color G with Z such that theextension G ∗ contains no C l , is at most D ( s + t ) . Proof of Lemma 12.
Let the vertex set of G be S ⊔ T with | S | = s and | T | = t . We applyinduction on s + t . By Lemma 11, | Z | := σ < l ( s + t ). The number of ways to color G isat most σ st . As long as s + t D/ l , we have σ st D st D ( s + t ) and this concludes the base case(s).For the induction step, we may henceforth assume s + t > D/ l , and the statement holdsfor all smaller values of s + t . Let us also assume without loss of generality that t s .Next, we deal with the case t s/ log s . Let D > l . Then s > ( s + t ) / > D/ l > l and the number of ways to color G is at most σ st (2 l ( s + t )) st s l ( s + t ))log s s ls )log s s ( s + t ) log D = D ( s + t ) . Therefore, we may assume that s/ log s < t s , and s > D/ l > s ( l ) so the conditionsof Theorem 18 hold. Let N G = ( z e ) e ∈ G be an edge-coloring of G using colors in Z . ByTheorem 18, such an edge-colored G will contain a subgraph G ′ that is either16 a rainbow colored K l, l , or • a Q -canonical K q, l , or • a monochromatic K q, l ,where | Q | = q = s ǫ . Claim 19 G ′ cannot be a rainbow colored K l, l .Proof of Claim 19. Suppose for a contradiction that G ′ = K l, l is rainbow colored and Z ′ is the set of colors used on G ′ . Then | Z ′ \ V ( G ′ ) | > l − l . Pick an edge of eachcolor in Z ′ \ V ( G ′ ) to obtain a strongly rainbow colored subgraph G ′′ of G ′ with | G ′′ | =16 l − l > (2 l − l . By Lemma 13, G ′′ contains a strongly rainbow colored cycle of length2 mod 2 l −
2. Lemma 14 now implies the existence of a strongly rainbow colored C l in G ′′ , which forms a linear C l in G ∗ , a contradiction.Let q = s ǫ and α be the number of edge-colorings of G that contain a Q -canonical subgraph G ′ which is a copy of K q, l and let β be the number of edge-colorings of G that containa monochromatic subgraph G ′ which is a copy of K q, l . We will prove that both α and β are at most (1 / D ( s + t ) and conclude by Claim 19 that the total number of colorings is atmost α + β D ( s + t ) as desired.Let the vertex set of G ′ = K q, l be Q ⊔ R , where Q ∈ (cid:0) Xq (cid:1) , R ∈ (cid:0) Y l (cid:1) and { X, Y } = { S, T } .Define a = | X | and b = | Y | so { a, b } = { s, t } . Our goal is to show that α (1 / D ( s + t ) . Recall that for each x ∈ Q , the edges in E ( x, R )all have the same color z x which is called a canonical color . Let Z c = { z x : x ∈ Q } be theset of all canonical colors. For each edge xy with x ∈ Q, y ∈ Y \ ( R ∪ Z c ), a color z xy = z x is called a free color . We will count the number of colorings of E ( Q, Y ), and then remove Q to apply the induction hypothesis. For each coloring N G , consider the following partitionof Y \ ( R ∪ Z c ) into two parts: Y = { y ∈ Y \ ( R ∪ Z c ) : E ( y, Q ) sees at most 11 l − } ,Y = { y ∈ Y \ ( R ∪ Z c ) : E ( y, Q ) sees at least 11 l free colors } . We claim that the length of strongly rainbow colored paths that lie between Q and Y isbounded. Claim 20
If there exists a strongly rainbow colored path P = P l − ⊂ E ( Q, Y ) with bothend-vertices u, v ∈ Q , then there exists a C l in G ∗ . roof of Claim 20. Clearly, P extends to a linear P l − in G ∗ . We may assume both z u , z v / ∈ V ( P ∗ ), where P ∗ = { e ∪ { z e } : e ∈ P } is the extension of P . Otherwise, suppose w.l.o.g. z u ∈ V ( P ∗ ), let y be the vertex next to u in P , let S y be of maximum size among sets { x ∈ Q : xy all colored by distinct free colors } . Since y ∈ Y , | S y | > l . Note that | V ( P ∗ ) | = 4 l − | V ( P ∗ ) ∩ Y | > l −
1, we have | S y \ V ( P ∗ ) | > l − (4 l − − ( l − > l . Since | V ( P ∗ ) | < l , E ( y, S y ) is rainbow, and G ′ is Q -canonical, there must be at least 4 l vertices in S y \ V ( P ∗ ) whose canonical color is notin V ( P ∗ ). Among these 4 l vertices there is at least one u ′ with z u ′ y / ∈ V ( P ∗ ). Replacing u by u ′ , we get a strongly rainbow colored path of length 2 l − z u / ∈ V ( P ∗ ).Now, Since | R | = 2 l , we can find a vertex y ∈ R such that y / ∈ { z e : e ∈ P } . Further, sinceboth z u , z v / ∈ V ( P ∗ ) and z u = z v , the set of edges P ∗ ∪ { uyz u , vyz v } forms a copy of C l in G ∗ .Thanks to this observation about strongly rainbow paths, we can bound the number ofcolorings on E ( Q, Y ) as follows. It is convenient to use the following notation. Definition 21
Given X ′ ⊂ X and Y ′ ⊂ Y , let E ( X ′ , Y ′ ) be the number of ways to colorthe edges in E ( X ′ , Y ′ ) . Claim 22 E ( Q, Y ) (2 l ) q · (32 l ) bq · ( qb ) lq · σ lq +8 l b . Proof of Claim 22.
By Claim 20, according to the length of the longest strongly rainbowcolored path starting at a vertex, Q can be partitioned into 2 l − F l − i =1 Q i , where Q i = { x ∈ Q : the longest strongly rainbow colored pathstarting at x and contained in E ( Q, Y ) has length i } . For each i , let q i = | Q i | . We now bound the number of colorings of the edges in E ( Q i , Y ).Firstly, for each x ∈ Q i , choose an i -path P x ⊂ E ( Q, Y ) starting at x and color it stronglyrainbow. The number of ways to choose and color these paths for all the vertices x ∈ Q i isat most (( qb ) ⌈ ( i +1) / ⌉ σ i ) q i ( qbσ ) iq i . Fix an x ∈ Q i . Partition Y into 3 parts depending on whether y is on the extension P ∗ x ofthe path starting at x , or the color of xy is on P ∗ x or else, i.e. Y = F j =1 Y ( j ) i,x , where Y (1) i,x = Y ∩ V ( P ∗ x ) ,Y (2) i,x = { y ∈ Y \ Y (1) i,x : z xy ∈ V ( P ∗ x ) } ,Y (3) i,x = Y \ ( Y (1) i,x ∪ Y (2) i,x ) . Depending on the part of Y that a vertex y lies in, we can get different restrictions on thecoloring of the edges in E ( y, Q i ). 18 If y ∈ Y (1) i,x , then z xy has as many as σ choices. Note that | P ∗ x | = 2 i + 1, and | Y (1) i,x | i + ⌈ i/ ⌉ i . This gives E ( x, Y (1) i,x ) σ i . • If y ∈ Y (2) i,x , then z xy ∈ V ( P ∗ x ), so there are at most 2 i + 1 choices for this color and E ( x, Y (2) i,x ) (2 i + 1) b . • Lastly, let | Y (3) i,x | = b i,x . If y ∈ Y (3) i,x , then xy extends P x into a strongly rainbowcolored path P ′ x = P x ∪ { xy } of length i + 1, which forces the edges x ′ y to be coloredby V ( P ′ x ∗ ) for each x ′ ∈ Q i \ V ( P ′ x ∗ ). Otherwise, the path P ′ x ∪ { x ′ y } is a stronglyrainbow colored path of length i + 2 starting at a vertex x ′ ∈ Q i , contradicting thedefinition of Q i . Therefore, z x ′ y has at most 2 i + 3 choices if x ′ ∈ Q i \ V ( P ′ x ∗ ). Puttingthis together, for each y ∈ Y (3) i,x , we have E ( Q i \ V ( P ′ x ∗ ) , y ) (2 i + 3) q i . Noticing that | Q i ∩ V ( P ′ x ∗ ) | i + 1 + ⌈ ( i + 1) / ⌉ i + 1, we have E ( Q i , y ) E ( Q i ∩ V ( P ′ x ∗ ) , y ) · E ( Q i \ V ( P ′ x ∗ ) , y ) σ i +1 (2 i + 3) q i . Hence the number of ways to color E ( x, Y ) ∪ E ( Q i , Y (3) i,x ) is at most2 b · σ i · (2 i + 1) b · σ (2 i +1) b i,x (2 i + 3) q i b i,x . The term 2 b arises above since Y (1) i,x has already been fixed before this step, so we just needto partition Y \ Y (1) i,x to get Y (2) i,x and Y (3) i,x .Now we remove x from Q i , Y (3) i,x from Y and repeat the above steps until we have the entire E ( Q i , Y ) colored. Note that P x ∈ Q i b i,x b , and that i l − i + 3 < l .We obtain E ( Q i , Y ) ( qbσ ) iq i Y x ∈ Q i b · σ i · (2 i + 1) b · σ (2 i +1) b i,x (2 i + 3) q i b i,x ( qbσ ) lq i Y x ∈ Q i b · σ l +4 lb i,x · (4 l ) b + q i b i,x ( qbσ ) lq i · bq i · σ lq i +4 lb · (4 l ) bq i + bq i = (32 l ) bq i · ( qb ) lq i · σ lq i +4 lb . Because P l − i =1 q i = q , taking the product over i ∈ [2 l − E ( Q, Y ) (2 l − q l − Y i =1 E ( Q i , Y ) (2 l − q l − Y i =1 (32 l ) bq i · ( qb ) lq i · σ lq i +4 lb (2 l ) q · (32 l ) bq · ( qb ) lq · σ lq +8 l b , where (2 l − q counts the number of partitions of Q into the Q i .19ince G ′ = E ( Q, R ) is Q -canonical, E ( Q, R ) σ q . As | Z c | q , E ( Q, Y ∩ Z c ) σ q . By definition of Y , E ( Q, Y ) ( σ l (11 l + 1) q ) b ( σ l (12 l ) q ) b . Therefore to color E ( Q, Y ), we need to first choose the subsets R and Z c ∩ Y of Y and thentake a partition to get Y and Y . We color each of E ( Q, R ) , E ( Q, Y ∩ Z c ) , E ( Q, Y ) and E ( Q, Y ). This gives E ( Q, Y ) b l b q b · E ( Q, R ) · E ( Q, Y ∩ Z c ) · E ( Q, Y ) · E ( Q, Y ) b l b q b · σ q · σ q · ( σ l (12 l ) q ) b · [(2 l ) q · (32 l ) bq · ( qb ) lq · σ lq +8 l b ]= b l b · (2 lb ) q · (384 l ) qb · ( qb ) lq · σ q + q +11 lb +6 lq +8 l b . Finally, we apply the induction hypothesis to count the number ways to color G = K s,t .Recall that q = s ǫ < s/ log s < t s , σ l ( s + t ) ls . There are two cases. • ( X, Y ) = (
S, T ) and ( a, b ) = ( s, t )Recall that we must first choose Q ⊂ X . E ( X, Y ) s q · E ( Q, Y ) · E ( X \ Q, Y ) s q · t l t · (2 lt ) q · (384 l ) qt · ( qt ) lq · σ q + q +11 lt +6 lq +8 l t · D ( s + t − q ) s q t l · t · (2 lt ) q · (384 l ) qt · ( qt ) lq · (4 ls ) q + q +11 lt +6 lq +8 l t · D ( s + t − q ) s q t l · t q · (2 /q (2 l ) /t l ) qt · ( qt ) lq · (4 ls ) l t · D − qt · D − qs + q · D ( s + t ) /q (2 l ) /t l (4 l ) l /q D ! qt · t l ( st ) q ( qt ) lq s l t · D − qs · D ( s + t ) /q (2 l ) /t l (4 l ) l /q D ! qt · q lq t l + q +2 lq s q +9 l t D qs · D ( s + t ) D ( s + t ) . To show the last inequality above, it is obvious that 2 /q (2 l ) /t l (4 l ) l /q < D for largeenough D , so we are left to show that 4 q lq t l + q +2 lq s q +9 l t < D qs for large D . Takinglogarithms, we havelog (cid:16) q lq t l + q +2 lq s q +9 l t (cid:17) = 2 + 2 lq log q + (2 l + q + 2 lq ) log t + ( q + 9 l t ) log s qs log D. D because qs has the dominating growth rate among all the aboveterms as q = s ǫ and s is large. • ( X, Y ) = (
T, S ) and ( a, b ) = ( t, s ) E ( X, Y ) t q · E ( Q, X ) · E ( Y \ Q, X ) t q · s l s · (2 ls ) q · (384 l ) qs · ( qs ) lq · σ q + q +11 ls +6 lq +8 l s · D ( s + t − q ) t q s l · s · (2 ls ) q · (384 l ) qs · ( qs ) lq · (4 ls ) q + q +11 ls +6 lq +8 l s · D ( s + t − q ) t q s l · s q · (2 /q (2 l ) /s l ) qs · ( qs ) lq · (4 ls ) l s · D − qs · D − qt + q · D ( s + t ) /q (2 l ) /s l (4 l ) l /q D ! qs · s l ( st ) q ( qs ) lq s l s · D − qt · D ( s + t ) /q (2 l ) /s l (4 l ) l /q D ! qs · q lq t q s l + q +2 lq +9 l t D qt · D ( s + t ) D ( s + t ) . Again, to show the last inequality above, it is clear that 2 /q (2 l ) /s l (4 l ) l /q < D for large D , so we are left to show that 4 q lq t q s l + q +2 lq +9 l s < D qs for large D . Takinglogarithms, we havelog (cid:16) q lq t q s l + q +2 lq +9 l s (cid:17) = 2 + 2 lq log q + q log t + (2 l + q + 2 lq + 9 l s ) log s qt log D. This holds for large D because qt has the dominating growth rate among all the aboveterms.In summary, the number of colorings of G such that there exists a G ′ ⊂ G that is a Q -canonical K q, l is α D ( s + t ) + 14 D ( s + t ) = 12 D ( s + t ) . Our goal is to show that β (1 / D ( s + t ) . Recall that the vertex set of G ′ = K q, l is Q ⊔ R ,where Q ∈ (cid:0) Xq (cid:1) and R ∈ (cid:0) Y l (cid:1) . The term canonical color now refers to the only color z c thatis used to color all edges of G ′ , and Z c = { z c } still means the set of canonical colors. A freecolor is a color that is not z c . As before we will count the number of colorings of E ( Q, Y ),and then remove Q to apply the induction hypothesis.Let Y = Y \ ( R ∪ Z c ). Similar to Claim 20, we claim that the length of a strongly rainbowcolored path between Q and Y is bounded. Claim 23
If there exists a strongly rainbow colored path P = P l − ⊂ E ( Q, Y ) with bothend-vertices u, v ∈ Q , then there exists a C l in G ∗ . roof of Claim 23. We observe that z c appears in the path or the color of the path at mostonce, as P is strongly rainbow. Hence, by the pigeonhole principle, there exists a sub-path P ′ of length 2 l − z c / ∈ V ( P ′∗ ) and both end-vertices u, v of P ′ are in Q .Now, Since | R | = 2 l , we can find two vertices y, y ′ ∈ R such that y, y ′ / ∈ { z e : e ∈ P ′ } . Thus,the edges P ′∗ ∪ { uyz c , vy ′ z c } yield a copy of C l in G ∗ .Again, we first use this claim to color E ( Q, Y ). Claim 24 E ( Q, Y ) (4 l ) q · (128 l ) qb · ( qb ) lq · σ lq +32 l b .Proof of Claim 24. The proof proceeds exactly the same as that of Claim 22, except that Q is partitioned into 4 l − F l − i =1 Q i . So in the calculation at the end, we have i l − i + 3 < l and E ( Q i , Y ) ( qbσ ) iq i Y x ∈ Q i b · σ i · (2 i + 1) b · σ (2 i +1) b i,x (2 i + 3) q i b i,x ( qbσ ) lq i Y x ∈ Q i b · σ l +8 lb i,x · (8 l ) b + q i b i,x ( qbσ ) lq i · bq i · σ lq i +8 lb · (8 l ) bq i + bq i (128 l ) bq i · ( qb ) lq i · σ lq i +8 lb . Again, note that P l − i =1 q i = q . Taking the product over i ∈ [4 l − E ( Q, Y ) (4 l − q l − Y i =1 E ( Q i , Y ) (4 l − q l − Y i =1 (128 l ) bq i · ( qb ) lq i · σ lq i +8 lb (4 l ) q · (128 l ) qb · ( qb ) lq · σ lq +32 l b , where (4 l − q counts the number of partitions of Q into the Q i .Similarly, to color E ( Q, Y ), we need to choose the subsets R and Y ∩ Z c , and what remainsis Y . Consequently, E ( Q, Y ) b l b · E ( Q, R ) · E ( Q, Y ∩ Z c ) · E ( Q, Y ) b l b · σ · σ q · [(4 l ) q · (128 l ) qb · ( qb ) lq · σ lq +32 l b ]= b l +1 (4 l ) q · (128 l ) qb · ( qb ) lq · σ q +12 lq +32 l b . We apply the induction hypothesis to count the number ways to color G = K s,t . Recallthat q = s ǫ < s/ log s < t s , σ l ( s + t ) ls . We split the calculation into two cases. • ( X, Y ) = (
S, T ) and ( a, b ) = ( s, t ) 22ecall that we must first choose Q ⊂ X . E ( X, Y ) s q · E ( Q, Y ) · E ( X \ Q, Y ) s q · t l +1 (4 l ) q · (128 l ) qt · ( qt ) lq · σ q +12 lq +32 l t · D ( s + t − q ) s q t l +1 · (4 l ) q · (128 l ) qt · ( qt ) lq · (4 ls ) q +12 lq +32 l t · D ( s + t − q ) s q t l +1 · ((4 l ) /t l ) qt · ( qt ) lq · (4 ls ) l t · D − qt · D − qs + q · D ( s + t ) (4 l ) /t l (4 l ) l /q D ! qt · t l +1 s q ( qt ) lq s l t · D − qs · D ( s + t ) (4 l ) /t l (4 l ) l /q D ! qt · q lq t l +1+4 lq s q +33 l t D qs · D ( s + t ) D ( s + t ) . To show the last inequality above, it is obvious to see that (4 l ) /t l (4 l ) l /q < D for large D , so we are left to show that 4 q lq t l +1+4 lq s q +33 l t < D qs for large D . Takinglogarithms, we havelog (cid:16) q lq t l +1+4 lq s q +33 l t (cid:17) = 2 + 4 lq log q + (2 l + 1 + 4 lq ) log t + ( q + 33 l t ) log s qs log D. This holds for large D because qs has dominating growth rate among all the terms above. • ( X, Y ) = (
T, S ) and ( a, b ) = ( t, s ) E ( X, Y ) t q · E ( Q, Y ) · E ( X \ Q, Y ) t q · s l +1 (4 l ) q · (128 l ) qs · ( qs ) lq · σ q +12 lq +32 l s · D ( s + t − q ) t q s l +1 · (4 l ) q · (128 l ) qs · ( qs ) lq · (4 ls ) q +12 lq +32 l s · D ( s + t − q ) t q s l +1 · ((4 l ) /s l ) qs · ( qs ) lq · (4 ls ) l s · D − qs · D − qt + q · D ( s + t ) (4 l ) /s l (4 l ) l /q D ! qs · s l +1 t q ( qs ) lq s l s · D − qt · D ( s + t ) (4 l ) /s l (4 l ) l /q D ! qs · q lq t q s l +1+4 lq +33 l t D qt · D ( s + t ) D ( s + t ) . To show the last inequality above, it is obvious to see that (4 l ) /s l (4 l ) l /q < D for large D , so we are left to show that 4 q lq t q s l +1+4 lq +33 l t < D qt for large D . Takinglogarithms, we havelog (cid:16) q lq t q s l +1+4 lq +33 l t (cid:17) = 2 + 4 lq log q + q log t + (2 l + 1 + 4 lq + 33 l t ) log s qt log D. D because qt has the dominating growth rate among all the termsabove.Finally, we have β D ( s + t ) + 14 D ( s + t ) = 12 D ( s + t ) , and the proof is complete. • A straightforward corollary of Theorem 2 is the very same result for hypergraph paths P k .Indeed, for the upper bound on Forb r ( n, P k ) one has to just observe that P k ⊂ C ⌈ ( k +1) / ⌉ ,while the lower bound is trivial. • Although we were unable to use hypergraph containers to prove Theorem 2 it wouldbe very interesting to give a new proof using containers. In particular, this would entailproving some supersaturation type results for this problem which may be of independentinterest. It would also likely yield some further results in the random setting which we havenot addressed. • The main open problem raised by our work is to solve the analogous question for larger r and for odd cycles (Conjecture 3).For r = 3, our method will not work for odd cycles as it relies on finding a bipartitestructure from which it is difficult to extract odd 3-uniform cycles (although this technicalhurdle could be overcome to solve the corresponding extremal problem in [37]).For larger r , our method does not work because the cost of decomposing a complete r -graphinto complete r -partite subgraphs is too large to remain an error term. More precisely, for r = 3, we implicitly applied Lemma 12 (in the proof of Lemma 8) to reduce the number ofways to color a graph to at most 2 O ( n ) instead of the trivial 2 O ( n log log n ) . But for r > O ( n r − (log n ) ( r − / ( r − ) which comes fromchoosing the colors for the copies of K s i : r − (see Section 3). This cannot be improved due toTheorem 6 and Lemma 10 each of which gives a bound that is sharp in order of magnitude.Consequently, even if we proved a version of Lemma 12 for r > r > • Another way to generalize the result of Morris-Saxton to hypergraphs is to consider similarenumeration questions when the underlying r -graph is linear, meaning that every two edgesshare at most one vertex. Here the extremal results have recently been proved in [20]and the formulas are similar to the case of graphs. The special case of this question forlinear triple systems without a C is related to the Ruzsa-Szem´eredi (6 ,
3) theorem and setswithout 3-term arithmetic progressions.
Acknowledgment.
We are very grateful to Rob Morris for clarifying some technical partsof the proof in [42] at the early stages of this project. After those discussions, we realized24hat the method of hypergraph containers would not apply easily to prove Theorem 2 andwe therefore developed new ideas. We are also grateful to Jozsef Balogh for providing uswith some pertinent references, and to Jie Han for pointing out that our proof of Theorem 2applies for r > k . References [1] N. Alon, J. Balogh, B. Bollob´as and R. Morris, The structure of almost all graphs ina hereditary property,
J. Combin. Theory, Ser. B
101 (2011), 85–110.[2] B. Bollob´as, Cycles modulo k , Bull. London Math. Soc.
Acta Informatica
23 (1986), 689–696.[4] J. Balogh, T. Bohman and D. Mubayi, Erd˝os-Ko-Rado in random hypergraphs,
Com-bin. Probab. Comput.
18 (2009), 629–646.[5] B. Bollob´as, B. Narayanan and A. Raigorodskii, On the stability of the Erd˝os-Ko-Radotheorem, arXiv:1408.1288.[6] J. Balogh, B. Bollob´as and M. Simonovits, On the number of graphs without forbiddensubgraph,
J. Combin. Theory, Ser. B (2004), 1–24.[7] , The typical structure of graphs without given excluded subgraphs, RandomStructures Algorithms
34 (2009), no. 3, 305–318.[8] , The fine structure of octahedron-free graphs,
J. Combin. Theory Ser. B
J. Combin. Theory, Ser. A
132 (2015), 224–245.[10] J. Balogh and D. Mubayi, Almost all triple systems with independent neighborhoodsare semi-bipartite,
J. Combin. Theory Ser. A
118 (2011), no. 4, 1494–1518.[11] , Almost all triangle-free triple systems are tripartite,
Combinatorica
32 (2012),no. 2, 143–169.[12] J. Balogh and A.Z. Wagner, On the number of union-free families, accepted,
Israel J.Math. [13] J. Balogh, R. Morris and W. Samotij, Independent sets in hypergraphs,
J. Amer. Math.Soc. (2015), 669–709.[14] J. Balogh and W. Samotij, The number of K m,m -free graphs, Combinatorica (2011),131–150.[15] , The number of K s,t -free graphs, J. London Math. Soc. (2011), 368–388.2516] J. A. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combin. TheorySer. B
16 (1974), 97–105.[17] G. Brightwell and S. Goodall, The number of partial orders of fixed width,
Order
Dis-crete Math.
201 (1999), no. 1-3, 53–80.[19] F.R.K. Chung, P. Erd˝os and J. Spencer, On the decomposition of graphs into completebipartite subgraphs
Studies in Pure Math. (1983), pp. 95–101 Mem. of P. Tur´an[20] C. Collier-Cartaino, N. Graber and T. Jiang, Linear Tur´an numbers of r -uniform linearcycles and related Ramsey numbers, arXiv:1404.5015v2, preprint, 2014.[21] D. Conlon and W.T. Gowers, Combinatorial theorems in sparse random sets, Ann. ofMath.
184 (2016), 367–454.[22] S. Das, T. Tran, Removal and Stability for Erd˝os-Ko-Rado
SIAM J. Discrete Math. k , J. Graph Theory 65 no. 3 (2010), 246–252.[25] P. Erd˝os, D.J. Kleitman and B.L. Rothschild, Asymptotic enumeration of K n -freegraphs, in Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Vol. II,pp. 19–27.
Atti dei Convegni Lincei , Accad. Naz. Lincei, Rome, 1976.[26] P. Erd˝os, P. Frankl and V. R¨odl, The asymptotic number of graphs not containinga fixed subgraph and a problem for hypergraphs having no exponent, Graphs andCombinatorics (1986), 113–121.[27] P. Erd˝os, On extremal problems of graphs and generalized graphs, Israel J. Math.
Bull. Amer. Math. Soc.
52 (12) (1946), 1087–1091.[29] Z. F¨uredi, T. Jiang, Hypergraph Tur´an numbers of linear cycles,
J. Combin. TheorySer. A
123 (1) (2014) 252–270.[30] Z. F¨uredi, T. Jiang and R. Seiver, Exact solution of the hypergraph Tur´an problem for k -uniform linear paths, Combinatorica
34 (2014), 299–322.[31] C. Hundack, H.J. Pr¨omel and Angelika Steger, Extremal graph problems for graphswith a color-critical vertex,
Combin. Probab. Comput.
DiscreteMath. (1982), 167–172. 2633] D. Kleitman and D. Wilson, On the number of graphs which lack small cycles,manuscript, 1996.[34] Ph.G. Kolaitis, H.J. Pr¨omel and B.L. Rothschild, K l +1 -free graphs: asymptotic struc-ture and a 0-1 law, Transactions of the Amer. Math. Soc.
Vol. 303, No. 2 (Oct., 1987),pp. 637–671[35] , Asymptotic enumeration of partial orders on a finite set,
Trans. Amer. Math.Soc.
205 (1975), 205–220.[36] T. K˝ov´ari, V. S´os and P. Tur´an, On a problem of K. Zarankiewicz,
Colloquium Math.
J. Combin. Theory, series A
129 (2015), 57–79.[38] , Tur´an problems and shadows II: trees,
J. Combin. Theory Ser. B
122 (2017),457–478.[39] , Tur´an problems and shadows III: expansions of graphs,
SIAM J. on DiscreteMath.
Vol. 29, No. 2 (2015), pp. 868–876[40] E. Lamken and B.L. Rothschild, The numbers of odd-cycle-free graphs, Finite andinfinite sets, Vol. I, II (Eger, 1981),
Colloq. Math. Soc. J´anos Bolyai vol. 37, North-Holland, Amsterdam, 1984, pp. 547–553.[41] C.-H. Liu, J. Ma, Cycle lengths and minimum degree of graphs, arXiv:1508.07912,preprint, 2015.[42] R. Morris and D. Saxton, The number of C l -free graphs, Adv. Math.
298 (2016),534–580.[43] D. Mubayi and G. Turan, Finding bipartite subgraphs efficiently,
Information Process-ing Letters
Volume 110, Issue 5, 1 (2010), 174–177.[44] B. Nagle and V. R¨odl, The asymptotic number of triple systems not containing a fixedone,
Discrete Math.
235 (2001), no. 1-3, 271–290,
Combinatorics (Prague, 1998).[45] B. Nagle, V. R¨odl and M. Schacht, Extremal hypergraph problems and the regularitymethod, Topics in Discrete Mathematics,
Algorithms Combin. (2006), 247–278.[46] D. Osthus, D. K¨uhn, T. Townsend, and Y. Zhao, On the structure of oriented graphsand digraphs with forbidden tournaments or cycles, arXiv:1404.6178, preprint, 2014.[47] Y. Person and M. Schacht, Almost all hypergraphs without Fano planes are bipar-tite, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algo-rithms, (2009), pp. 217–226.[48] H.J. Pr¨omel and A. Steger, The asymptotic number of graphs not containing a fixedcolor-critical subgraph,
Combinatorica
12 (1992), no. 4, 463–473.2749] R.W. Robinson, Counting labeled acyclic digraphs,
New directions in the theory ofgraphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971) (1973),pp. 239–273.[50] D. Saxton and A. Thomason, Hypergraph containers,
Inventiones Math. (2015),1–68.[51] R.P. Stanley, Acyclic orientations of graphs,
Discrete Math. l mod k , Elec.J. of Combin. (2015), k , J. Graph Theory
Matematikai ´es Fizikai Lapok (inHungarian) 48 (1941), 436–452.[55] Z. Tuza, Covering of graphs by complete bipartite subgraphs; complexity of 0-1 matri-ces,
Combinatorica4