aa r X i v : . [ m a t h . C O ] M a y Tur´an numbers of bipartite subdivisions
Tao Jiang ∗ Yu Qiu † May 28, 2019
Abstract
Given a graph H , the Tur´an number ex( n, H ) is the largest number of edges in an H -freegraph on n vertices. We make progress on a recent conjecture of Conlon, Janzer, and Lee [7] onthe Tur´an numbers of bipartite graphs, which in turn yields further progress on a conjecture ofErd˝os and Simonovits [8].Let s, t, k ≥ K ks,t denote the graph obtained from the complete bipartitegraph K s,t by replacing each edge uv in it with a path of length k between u and v such that the st replacing paths are internally disjoint. It follows from a general theorem of Bukh and Conlon[2] that ex( n, K ks,t ) = Ω( n k − sk ). Conlon, Janzer, and Lee [7] recently conjectured that for anyintegers s, t, k ≥
2, ex( n, K ks,t ) = O ( n k − sk ). Among many other things, they settled the k = 2case of their conjecture. As the main result of this paper, we prove their conjecture for k = 3 , Tur´an exponents : rationals r ∈ (1 , H with ex( n, H ) = Θ( n r ), adding to the lists recentlyobtained by Jiang, Ma, Yepremyan [23], by Kang, Kim, Liu [24], and by Conlon, Janzer, Lee [7].Our method builds on an extension of the Conlon-Janzer-Lee method. We also note that theextended method also gives a weaker version of the Conlon-Janzer-Lee conjecture for all k ≥ Given a family H of graphs, the Tur´an number ex ( n, H ) is the largest number of edges in an n -vertexgraph that does not contain any member of H . If H consists of a single graph H , we write ex ( n, H )for ex ( n, { H } ). Let p = min { χ ( H ) − H ∈ H} , where χ ( H ) denotes the chromatic number of H . The celebrated Erd˝os-Stone-Simonovits theorem asserts that ex( n, H ) = (1 − p + o (1)) (cid:0) n (cid:1) . Thisdetermines the function for all families that do not contain a bipartite member. When H containsa bipartite graph, the problem is generally wide-open, with many intriguing conjectures. One ofthese, known as the Tur´an exponent conjecture , was made by Erd˝os and Simonovits [8] that assertsthat for any rational r ∈ (1 ,
2) there exists a bipartite graph H such that ex( n, H ) = Θ( n r ). Wecall a rational r for which the Erd˝os-Simonovits conjecture holds a Tur´an exponent . In a recentbreakthrough, Bukh and Conlon [2] have proved that for any rational number r ∈ (1 ,
2) there exists ∗ Department of Mathematics, Miami University, Oxford, OH 45056, USA. E-mail: [email protected]. † School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China. Email:[email protected]. Research supported by China Scholarship Council.
Key Words : Tur´an number, Tur´an exponent, extremal function, subdivision iang, Qiu: On Tur´an numbers of bipartite subdivisions
2a finite family H of graphs such that ex( n, H ) = Θ( n r ). On the other hand, the original conjectureof Erd˝os and Simonovits concerning single bipartite graphs is still generally open. Until recently,it was only known to be true for r = 1 + 1 /k and r = 2 − /k where k ≥ H , and aninteger k ≥
2, let H k denote the graph obtained by replacing each edge uv of H with a path of length k between u and v so that the e ( H ) replacing paths are internally vertex disjoint. The Tur´an numberof H k is studied in [19] and [21], based on earlier work in [26]. Recently, significant progresses onthe problem have been made in [6], [18], and [7]. Let s, t, k ≥ K s,t denote the complete bipartite graph with part sizes s and t . Let K ks,t = ( K s,t ) k . It follows fromthe above mentioned breakthrough work of Bukh and Conlon [2] that ex( n, K ks,t ) = Ω( n k − sk ).Conlon, Janzer, and Lee [7] recently made the following conjecture on a matching upper bound. Conjecture 1.1 [7]
For any integers s, t, k ≥ , ex( n, K ks,t ) = O ( n k − sk ) . In [7], among many other things, Conlon, Janzer and Lee settled the k = 2 case of Conjecture1.1, showing that ex( n, K s,t ) = O ( n − s ). In this paper, we prove their conjecture for k = 3 , Theorem 1.2
For any integers s, t ≥ and k ∈ { , } , ex( n, K ks,t ) = O ( n k − sk ) . We remark that our theorem together with the theorem of Bukh and Conlon also yields infinitelymany new
Tur´an exponents : namely those of the form 1 + k − sk , where s ≥ k ∈ { , } . The majority of the rest of the paper is devoted to the proof of our main result: Theorem1.2. We then conclude with some observations in the concluding remarks. As is often the case in the study of bipartite Tur´an problems, our problem may be reduced to thesetting in which the host graph is almost regular. Specifically, given a positive integer K , we saythat a graph G is K -almost-regular if ∆( G ) ≤ K · δ ( G ).The following lemma can be found in [21], which is a slight adaption of the regularization lemmaof Erd˝os and Simnovits [11]. Another recent adaption of this can be found in [7]. Lemma 2.1 ([21]] Proposition 2.7)
Let < ǫ < and c ≥ . There exists n = n ( ǫ ) > such thatthe following holds for all n ≥ n . If G is a graph on n vertices with e ( G ) ≥ cn ǫ , then G containsa K -almost regular subgraph G ′ on m ≥ n ǫ − ǫ ǫ vertices such that e ( G ′ ) ≥ c m ǫ and K = 20 · ǫ +1 . For most of the rest of the paper we will always assume our host graph G to be almost regular.Then in the main proof we apply Lemma 2.1 on general host graphs. iang, Qiu: On Tur´an numbers of bipartite subdivisions In this section, we present one of the main ingredients used by Conlon, Janzer, and Lee [7]. To makeour presentation consistent with the rest of our paper, we present their results using our notationand terminology.
Definition 3.1 (Definition 6.2 of [7]) Let L be an integer. Define the function f ( ℓ, L ) for 0 ≤ ℓ ≤ k recursively by setting f (0 , L ) = 1 , f (1 , L ) = L and, for 2 ≤ ℓ ≤ k , f ( ℓ, L ) = 1 + f ( ℓ − ( ℓ − max ≤ i ≤ ℓ − f ( i, L ) f ( ℓ − i, L ) . Definition 3.2
Let L be fixed. Let G be a graph. For each ℓ ≥
1, a path P of length ℓ in G withendpoints x, y is called ℓ -heavy if there are more than f ( ℓ, L ) distinct x, y -paths of length ℓ in G andis called ℓ -light otherwise. A path P of length ℓ in G is called ℓ -critical if it is ℓ -heavy but for each j < ℓ each subpath of length j is j -light. Since the length of a given path is fixed, we may drop theprefix and use terms heavy, light, critical directly.In [7], lights paths are called good paths and critical paths are called admissible paths . The followinglemma is implied by Lemma 6.8 and Corollary 6.9 of [7] since their forbidden subgraph H is asupergraph of K ks,t . Lemma 3.3
Let G be a K ks,t -free K -almost-regular graph on n vertices with minimum degree δ .Then provided that L is sufficiently large compared to s, t, k and K , for any ≤ ℓ ≤ k , the numberof ℓ -critical paths is at most n Kδ ) ℓ f ( ℓ − ,L ) . Lemma 3.3 roughly says if a graph has many short critical paths, then we can easily build a copyof K ks,t . In the next subsection, we develop analogous statements for other critical substructures. A non-path spider is a tree with exactly one vertex w of degree at least three, called the center .Paths from the center to the leaves are called legs . A spider in which all legs have length h is called a balanced spider of height h . In this section, whenever we discuss a non-path spider T with m legs, wealways fix a particular labelling of its leaves as v , . . . , v m . For each i ∈ [ m ], let ℓ i be distance fromthe center w of T to v i . We call T a spider with leaf vector ( v , . . . , v m ) and length vector ( ℓ , . . . , ℓ m ). Definition 3.4
Let s, k ≥ G be a graph. Let ~ℓ := ( ℓ , . . . , ℓ s ) be a vector of s positive integers, each of which is at most k . We say that a vertex ordered tuple ( v , . . . , v s ) in G is ( ℓ , . . . , ℓ s )- strong if G contains at least ( sk ) sk − ℓ · f ( k, L ) internally vertex-disjoint spiders withleaf vector ( v , . . . , v s ) and length vector ( ℓ , . . . , ℓ s ), where ℓ = ℓ + · · · + ℓ s . A spider with leafvector ( v , . . . , v s ) and length vector ( ℓ , . . . , ℓ s ) is called ( ℓ , . . . , ℓ s )- strong if the tuple ( v , . . . , v s ) is( ℓ , . . . , ℓ s )-strong. Since the length vector of any spider is fixed, whenever we say a spider is strong,it is understood that it is strong relative to its length vector. iang, Qiu: On Tur´an numbers of bipartite subdivisions Lemma 3.5
Let G be a K -almost-regular graph with minimum degree δ . Let x be a vertex. Let C bea family of at least αδ h distinct paths of length h with one end x and another end in a set S . Then C contains a subfamily D of more than ( α/hK h − ) δ paths which are vertex-disjoint outside { x } .Proof. Let
D ⊆ C be a maximal subfamily of paths that are vertex disjoint outside { x } . Let W be the set of vertices contained in these paths besides x . Then | W | = h |D| . By the maximality of D each member of C must pass through x and some vertex in W . Since G has maximum degree atmost Kδ , there can be at most | W | ( Kδ ) h − such paths. Hence |C| ≤ | W | ( Kδ ) h − . Since |C| ≥ αδ h and | W | = h |D| , we have |D| > ( α/hK h − ) δ . Lemma 3.6
Let G be a K -almost-regular graph with minimum degree δ . Let x be a vertex. Let C bea family of at least αδ h distinct paths of length h with one end x and another end in a set S . Let F be the subgraph of G formed by taking the union of paths in C . For each i ∈ [ h ] there exists a vertex x i and a balanced spider of height i with center x i and leaves in S and has at least ( α/hK h − ) δ legs.Furthermore, if i = h , then x i = x .Proof. Since G has maximum degree at most Kδ , there are at most ( Kδ ) h − i distinct paths of length h − i starting at x . So there is a path Q of length h − i starting at x and ending at some vertex x i that is the initial segment of at least |C| / ( Kδ ) h − i ≥ ( α/K h − i ) δ i members of C . If i = h − i , then x i = x . Let C i denote the subfamily consisting of these members. Then { P − V ( Q ) : P ∈ C ′ } is afamily of |C ′ | distinct paths of length i each of which starts at x i and ends in S . By Lemma 3.5, C ′ contains a subfamily of size at least [( α/K h − i ) /iK i − ] δ ≥ ( α/hK h − ) δ which are vertex-disjointoutside { x i } . The claim holds.An s -uniform hypergraph F is called s -partite if there exists a partition of V ( F ) into A , . . . , A s such that each edge contains one vertex from each A i . We call the A i ’s the parts. Lemma 3.7
Let F be an s -partite s -graph with parts A , . . . , A s . Suppose that |F | > α | A | · · · | A s | ,where α > . Let i ∈ [ s ] . Then there exists a subgraph F ′ such that |F ′ | ≥ (1 / |F | and for each v ∈ A i ∩ V ( F ′ ) , d F ′ ( v ) > ( α/ Q j ∈ [ s ] \{ i } | A j | .Proof. Let us call a vertex v ∈ A i it i -bad if its degree in the remaining graph is at most( α/ Q j = i | A j | . As long as there exists an i -bad vertex for some i ∈ [ s ], we remove all the edgescontaining that vertex. Let F ′ be the remaining subgraph. Then at most ( α/ Q si =1 | A i | edges areremoved in the process. So |F ′ | > (1 / |F | . Clearly F ′ satisfies the degree requirement.Note that one could easily modify Lemma 1.8 to apply to all parts. But it suffices our purposes.The following lemma provides one of the main ingredients of our proofs of the main results. Lemma 3.8
Let K ≥ and integers k, s, t ≥ be fixed. Then provided that L is sufficiently largecompared to s, t, k and K , for any β > there exists δ such that the following holds. Suppose that G is an K ks,t -free K -almost-regular graph n vertices with minimum degree δ ≥ δ . If ℓ , . . . , ℓ s arepositive integers satisfying that ∀ i ∈ [ s ] , ℓ i ≥ k/ and that ∀ ≤ i < j ≤ s, ℓ i + ℓ j ≥ k + 1 , then thenumber of tuples ( w, v , . . . , v s ) such that there is an ( ℓ , . . . , ℓ s ) -strong spider with center w and leafvector ( v , . . . , v s ) is at most βnδ ℓ , where ℓ = ℓ + · · · + ℓ s . iang, Qiu: On Tur´an numbers of bipartite subdivisions Proof.
For each vertex w in G , let H w denote the family of tuples ( v , . . . , v s ) such that there isan ( ℓ , . . . , ℓ s )-strong spider with center w and leaf vector ( v , . . . , v s ). Suppose for contradictionthat there exists more than βnδ ℓ tuples ( w, v , . . . , v s ) such that there is an ( ℓ , . . . , ℓ s )-strong spiderwith center w and leaf vector ( v , . . . , v s ). Then by the pigeonhole principle, there exists a vertex w such that |H w | > βδ ℓ . Let us fix such a w . For each ( v , . . . , v s ) ∈ H w , by definition, we may fix a( ℓ , . . . , ℓ s )-strong spider T ( v , . . . , v s ) with leaf vector ( v , . . . , v s ). For each i , we call the path in T ( v , . . . , v s ) from w to v i its i -th leg.Randomly and independently color vertices of G with colors 1 , . . . , s with each vertex receivingeach color with probability 1 /s . For each s -tuple ( v , . . . , v s ) ∈ H w , we call it good if for each i ∈ [ s ]all the vertices on the i -th leg of T ( v , . . . , v s ) except w are colored i . Since T ( v , . . . , v s ) − { w } has ℓ vertices, the probability of ( v , . . . , v s ) being good is 1 /s ℓ . Hence, there exists a coloring c such thatthe following family F w = { ( v , . . . , v s ) ∈ H w : ( v , . . . , v s ) is good } satisfies |F w | ≥ |H w | /s ℓ > ( β/s ℓ ) δ ℓ . (1)Let us fix such a coloring c . For each i ∈ [ s ], let A i = { v ∈ V ( F w ) : c ( v ) = i } . Then F w is an s -partite s -graph with parts A , . . . , A s . By our assumption, for each i ∈ [ s ] and each v ∈ A i there is an ( ℓ , . . . , ℓ s )-strong spider with center w where v plays the role of the i -th vertex inthe leaf vector. Furthermore, all the vertices on the i -th leg, except w , are colored i under c . Since G has maximum degree at most Kδ , we have ∀ i ∈ [ s ] , | A i | ≤ ( Kδ ) ℓ i . (2)Let α = βs ℓ K ℓ . For each i ∈ [ s ], let α i = βs ℓ K ℓ − ℓi . By (1) and (2), we have |F w | > α | A | . . . | A s | and ∀ i ∈ [ s ] , | A i | ≥ |F w | / Y j = i | A j | ≥ α i δ ℓ i . (3)Now, we may assume without loss of generality that ℓ ≤ ℓ ≤ · · · ≤ ℓ s . First, let us observethat if ℓ = ℓ = · · · = ℓ s = k , then we may take any ( k, . . . , k )-strong tuple ( v , · · · , v s ). By thedefinition of strong tuples, there are at least f ( k, L ) internally vertex-disjoint spiders with leaf vector( v , . . . , v s ). It is easy to see that the union of any t of these spiders form a copy of K ks,t , contradicting G being K ks,t -free. Hence, we may assume that ℓ < k . For each i ∈ [ s ], let m i = k − ℓ i . By ourassumption, ∀ i ∈ [ s ] , ℓ i ≥ k/ ∀ i, j ∈ [ s ] , ℓ i + ℓ j > k . This implies that m ≤ ℓ and ∀ ≤ i ≤ s, m i < ℓ i . Let q = max { i : ℓ i < k } . Then 1 ≤ q ≤ s . By Lemma 3.7, F w contains a subgraph F such that |F | > (1 / |F w | > ( α/ | A | · · · | A s | . (4) iang, Qiu: On Tur´an numbers of bipartite subdivisions ∀ v ∈ A ′ := A ∩ V ( F ) , d F ( v ) ≥ ( α/ Y j =1 | A j | . (5)By (4) and (3), we have | A ′ | ≥ ( α/ | A | ≥ (1 / αα δ ℓ . (6)For each v ∈ A ′ there is an edge of F containing it, which in particular, by our earlier discussion,implies that there is a path P v of length ℓ from w to v , all of which except w are colored i by c . Let β = (1 / αα ℓ K ℓ − . By Lemma 3.6, there exists a vertex z and a balanced spider S of height m with center at z andleaf set B ⊆ A ′ such that β δ ≤ | B | ≤ δ. Note that if m = ℓ , then z = w . If m < ℓ , then z = w . Also, all the vertices in S , exceptpossibly w , have color 1 in c . Since B ⊆ A ′ , by (5) ∀ v ∈ B , d F ( v ) ≥ ( α/ Y j =1 | A j | . (7)In general, let 1 ≤ i ≤ q − F , . . . , F i and B , . . . , B i such that |F i | ≥ ( α/ i ) | B | · · · | B i − || A i | · · · | A s | and β i δ ≤ | B i | ≤ δ, where β i = (1 / i ) αα i ℓ i K ℓ i − . Furthermore, suppose ∀ v ∈ B i , d F i ( v ) ≥ ( α/ i ) | B | · · · | B i − || A i +1 | · · · | A s | . (8)Also, suppose that there are distinct vertices z , . . . , z i such that for each j ∈ [ i ], there is abalanced spider S j of height m j with center z j and leaf set B j , all of whose vertices except possibly w lie in color class j of c . Also, suppose that z , . . . , z i = w and z = w if and only if ℓ = m . Now,let H i +1 be the subgraph of F i induced by B ∪ · · · ∪ B i ∪ A i +1 ∪ · · · ∪ A s . By (8), |H i +1 | ≥ ( α/ i ) | B | · · · | B i || A i +1 | · · · | A s | . (9)By Lemma 3.7, H i +1 contains a subgraph F i +1 such that |F i +1 | ≥ (1 / |H i +1 | ≥ ( α/ i +1 ) | B | · · · | B i || A i +1 | · · · | A s | . (10)and ∀ v ∈ A ′ i +1 := A i +1 ∩ V ( F i +1 ) , d F i +1 ( v ) ≥ ( α/ i ) | B | · · · | B i || A i +2 | · · · | A s | . (11) iang, Qiu: On Tur´an numbers of bipartite subdivisions | A ′ i +1 | ≥ ( α/ i +1 ) | A i +1 | ≥ (1 / i +1 ) αα i +1 δ ℓ i +1 . (12)As before, for each v ∈ A ′ i +1 there is a path P v of length ℓ i +1 from w to v , all of which except w have color i + 1 in c . Let β i +1 = (1 / i +1 ) αα i +1 ℓ i +1 K ℓ i +1 − . By Lemma 3.6, there exists a vertex z i +1 and a balanced spider S i +1 of height m i +1 with center z i +1 and leaf set B i +1 ⊆ A ′ i +1 such that β i +1 δ ≤ | B i +1 | ≤ δ. Furthermore, since m i +1 < ℓ i +1 , we have z i +1 = w . Also, all the vertices in S i +1 lie in color class i + 1 of c . Since B i +1 ⊆ A ′ i +1 , by (11) ∀ v ∈ B i +1 , d F i +1 ( v ) ≥ ( α/ i +1 ) | B | · · · | B i || A i +2 | · · · | A s | . This allows to define F , . . . , F q , B , . . . , B q , and z , . . . , z q . Now, we claim that we can find acopy of K ks,t in G , which would give us a contradiction. To find such a copy, we consider two subcases. Case 1. q = s .By our assumption, F s is an s -partite s -graph with parts B , . . . , B s , where |F s | ≥ ( α/ s ) | B | · · · | B s | and ∀ i ∈ [ s ] , β i δ ≤ | B i | ≤ δ, where β i = (1 / s ) αα s ℓ s K ℓ s − . Let M be a maximum matching in F s . Then the maximality of M implies that every edge of F s contains some vertex in V ( M ). On the other hand, since F s is s -partite and each part has sizeat most δ , each vertex is contained in at most δ s − edges. Hence |F s | ≤ | V ( M ) | · δ s − = s |M| δ s − . Hence by the above lower bounds on |F s | and | B | , . . . , | B s | , we have |M| ≥ |F s | /sδ s − ≥ ( αβ · · · β s / s ) δ ≫ t, for sufficiently δ (as δ ≥ δ ). Let M ′ be a set of t edges in M . Suppose M ′ = { e , . . . , e t } . For each i ∈ [ t ], suppose e i = ( v i , v i , . . . , v is ), where ∀ j ∈ [ s ] , v ij ∈ B j . For each j ∈ [ s ], let Z j be the sub-spiderof S j obtained by keeping only the t paths from z j to V ( M ′ ) ∩ B j . Since vertices in Z − { w } havecolor 1 and for each 2 ≤ j ≤ s , vertices in Z j have color j , Z . . . , Z t are vertex-disjoint.By the definition of F s ⊆ H w , for each i ∈ [ t ], ( v i , . . . , v is ) is a strong ( ℓ , . . . , ℓ s )-tuple andhence there are f ( k, L ) internally vertex-disjoint spiders with leaf vector ( v i , . . . , v is ) and length iang, Qiu: On Tur´an numbers of bipartite subdivisions ℓ , . . . , ℓ s ). Since f ( k, L ) ≫ | V ( K ks,t ) | , we can greedily find t vertex disjoint spiders T , . . . T t such that for each i ∈ [ t ], T i has leaf vector ( v i , . . . , v is ) and length vector ( ℓ , . . . , ℓ s ) and that V ( T i ) \ { v i , v i , . . . , v is } is disjoint from S sj =1 V ( Z j ). Now ( S ti =1 T i ) ∪ ( S sj =1 Z j ) forms a copy of K ks,t ,contradicting G being K ks,t -free. Case 2. q < s .Since |F s | ≥ ( α/ s ) | B | · · · | B s | , by averaging, there exists a tuple ( z q +1 , · · · , z s ) ∈ B q +1 × · · · × B s that is contained in at least ( α/ s ) | B | · · · | B q | of the edges of F s . Let F ∗ = { e \ { z q +1 , . . . , z s } : { z q +1 , . . . , z s } ⊆ e ∈ F s } . As in Case 1, for sufficiently large n , F ∗ contains a matching M = { e , . . . , e t } of size t .For each i ∈ [ t ], suppose e i = ( v i , v i , . . . , v iq ), where ∀ j ∈ [ q ] , v ij ∈ B j . For each j ∈ [ q ], let Z j bethe sub-spider of S j obtained by keeping only the t paths from z j to V ( M ) ∩ B j . Since vertices in Z −{ w } have color 1 and for each 2 ≤ j ≤ s , vertices in Z j have color j , Z . . . , Z t are vertex-disjoint.By definition, for each i ∈ [ t ], ( v i , . . . , v iq , z q +1 , . . . , z s ) is a strong ( ℓ , . . . , ℓ s )-tuple and hence thereare f ( k, L ) internally vertex-disjoint spiders with leaf vector ( v i , . . . , v is ) and length vector ( ℓ , . . . , ℓ s ).Since f ( k, L ) ≫ | V ( K ks,t ) | , we can greedily find t spiders T , . . . T t such that for each i ∈ [ t ], T i hasleaf vector ( v i , . . . , v iq , z q +1 , . . . , z s ) and length vector ( ℓ , . . . , ℓ s ) and that V ( T i ) \ { z q +1 , . . . , z s } arepairwise disjoint over different i and that V ( T i ) \ { v i , v i , . . . , v is } is disjoint from S qj =1 V ( Z j ) for each i ∈ [ t ]. Now ( S ti =1 T i ) ∪ ( S sj =1 Z j ) forms a copy of K ks,t , contradicting G being K ks,t -free.From Lemma 3.8, we immediately obtain the following. Corollary 3.9
Let K ≥ and integers k, s, t ≥ be fixed. Then provided that L is sufficiently largecompared to s, t, k and K , for any β > there exists δ such that the following holds. Suppose that G is an K ks,t -free K -almost-regular graph n vertices with minimum degree δ ≥ δ . Suppose ℓ , . . . , ℓ s are positive integers satisfying that ∀ i ∈ [ s ] , ℓ i ≥ k/ and that ∀ ≤ i < j ≤ s, ℓ i + ℓ j ≥ k + 1 .Let ℓ = ℓ + · · · + ℓ s . Let F denote the family of all the balanced s -legged spiders in G of height k that contain a ( ℓ , . . . , ℓ s ) -strong sub-spider but contain no critical path of length at most k . Then |F | ≤ [ f ( k, L )] s · βnδ ℓ .Proof. Let F ∈ F . By Definition 3.2, since F contains no critical paths of length at most k , it alsodoes not contain any heavy paths of length at most k . By Lemma 3.8, there are at most βnδ ℓ tuples( w, ℓ , . . . , ℓ s ) such that there is a member of F that has w as the center and ( ℓ , . . . , ℓ s ) as the leafvector. Each such tuple corresponds to at most [ f ( k, L )] s different members of F , since for each i ,there are at most f ( k, L ) light paths of length k in G between w and v i . (1 , k, . . . , k ) -case In this section, we prove a second crucial ingredient (Lemma 3.12 below) which complements Lemma3.8. First we need a lemma (Lemma 3.11 below), which is a slight adaption of [21] Lemma 2.4. Givena u, w -path P in a graph and vertices x, y on P , we let P [ x, y ] denote the portion of P from x to y . iang, Qiu: On Tur´an numbers of bipartite subdivisions Lemma 3.10
Let m, k be positive integers. Let A , A , . . . , A k be disjoint sets of vertices, where A = { z } and | A k | ≥ m k . For each w ∈ A k , let P w be a fixed z, w -path of length of k that containsexactly one vertex of each A i . Then there exists a vertex x ∈ A k − j for some j ∈ [ k ] and m vertices w , . . . , w m in A k such that { P w i [ x, w i ] : i ∈ [ m ] } is a family of paths of length j every two of whichshare only x as a common vertex.Proof. We prove the statement by induction on k . The case of k = 1 is trivial. Assume that k ≥ k −
1. For each w ∈ W , let f ( w ) denote the vertex on P w that precedes w . Then we may view S := { f ( w ) : w ∈ W } as a multi-set of size | W | . If some vertex x in S hasmultiplicity m in S , then there exist w , . . . , w m such that f ( w ) = f ( w ) = · · · = f ( w m ) = x andthe claim holds with j = 1. Hence, we may assume that each vertex in S is the image of fewer than m vertices in A k under f . In particular, we have | S | ≥ | A k | /m = m k − . For each y ∈ S , let h ( y ) be an arbitrary pre-image of y under f . Then h is an injection from S to A k .For each y ∈ S let Q y = P h ( y ) [ z, y ]. Since | S | ≥ m k − , by the induction hypothesis, there exist somevertex x ∈ A k − − j for some j ∈ [ k −
1] and m vertices y , . . . y m in S such that { Q y i [ x, y i ] : i ∈ [ m ] } is a family of paths of length j every two of which share only x as a common vertex. For i ∈ [ m ], Q y i [ x, y i ] ∪ y i h ( y i ) = P h ( y i ) [ x, h ( y i )] form a family of paths that satisfy the statement. Lemma 3.11
Let m and k be integers. Let z be a vertex and W is a set not containing z . For each w ∈ W , let P w be a z, w -path of length k and let F = { P w : w ∈ W } . If | W | ≥ ( mk ) k , then for some j ∈ [ k ] there exist a vertex x and m vertices w , . . . , w m in W such that { P w i [ x, w i ] : i ∈ [ m ] } is afamily of paths of length j every two of which share only x as a common vertex.Proof. Let us randomly and independently color the vertices in S w ∈ W V ( P w ) − { z } using 1 , . . . , k with each color chosen with probability 1 /k . Let us call a P w ∈ F good if for each i ∈ [ k ] the vertexon P w at distance i from z is colored i . The probability of any P w being good is (1 /k ) k . Hencethere exists a coloring for which the number of good P w ’s is at least ( mk ) k /k k = m k . Now the claimfollows immediately from Lemma 3.10. Lemma 3.12
Let K ≥ and integers k, s, t ≥ be fixed. Then provided that L is sufficiently largecompared to s, t, k and K , for any γ > there exist n , C > such that the following holds. Supposethat G is an K ks,t -free K -almost-regular graph n ≥ n vertices with minimum degree δ ≥ Cn k − sk . Let F denote the family of all the balanced s -legged spiders of height k in G that contain a (1 , k, . . . , k ) -strong s -legged spider but do not contain any critical paths of length at most k or any ( ℓ , . . . , ℓ s ) -strong sub-spider for any ( ℓ , . . . , ℓ s ) = (1 , k, . . . , k ) . Then |F | ≤ γnδ sk .Proof. Suppose to the contrary that |F | ≥ γnδ ks . We derive a contradiction. First we do somecleaning. Let c > skK ) sk c = γ and let ∂ ( F ) = { T : T is a tree on at most ks vertices and ∃ F ∈ F , E ( T ) ⊆ E ( F ) } iang, Qiu: On Tur´an numbers of bipartite subdivisions T ∈ ∂ ( F ) such that there are fewer than cδ · ( Kδ ) sk − e ( T ) − = cδ ( Kδ ) sk −| T | members of F that contain T , we delete all these members from F ; otherwise, terminate. Let F ′ denote the remaining subfamily of F .For each j ∈ [ sk ] let ∂ j ( F ) = { T ∈ ∂ ( F ) : | T | = j } . Let T j denote the set of all labelled trees on[ j ]. By Cayley’s formula |T j | ≤ j j − < j j . For each member T ∈ T j , there are at most n · ( Kδ ) j − copies of T in G , since G has maximum degree at most Kδ . Hence | ∂ j ( F ) | ≤ j j · n ( Kδ ) j − . Onthe other hand, for each T ∈ ∂ j ( F ), by rule, we have deleted fewer than cδ · ( Kδ ) sk − j membersfrom F that contain T . Thus the total number of members we have deleted from F is less than P skj =1 j j · n ( Kδ ) j − · cδ ( Kδ ) sk − j ≤ ( sk ) sk cnδ ( Kδ ) sk − ≤ γnδ ks . Hence |F ′ | ≥ γnδ ks − γnδ ks = 34 γnδ ks , and by the definition of F ′ ∀ T ∈ ∂ ( F ′ ) there are at least cδ ( Kδ ) sk −| T | members of F ′ that contain T . (13)Given an ( s − ~a = ( a , . . . , a s − ) of vertices in G , let L ~a denote the subfamily of membersof F ′ that contain a , . . . , a s − as leaves. For each F ∈ L ~a , let w ( F ) denote the center of F and let u ( F ) denote the neighbor of w ( F ) on the path from w ( F ) to the remaining leaf z . For each F ∈ L ~a ,let F | ~a denote the subtree obtained from F by replacing the w ( F ) , z - path in it with w ( F ) u ( F ). If F | ~a is a (1 , k, . . . , k )-strong spider with center w ( F ) and leaf vector ( u ( F ) , a , . . . , a s − ) then we saythat F ~a is good . For each ( s − ~a let F ~a = { F ∈ F ′ : F | ~a is good } , and let H ~a = { w ( F ) u ( F ) : F ∈ F ~a } . Furthermore, let W ~a = { w ( F ) : F ∈ F ~a } and U ~a = { u ( F ) : F ∈ F ~a } . Since G is bipartite, we have W ~a ∩ U ~a = ∅ . Hence H ~a is bipartite with parts W ~a and U ~a . Observethat by definition, ∀ u ∈ U ~a , there is a (1 , k, . . . , k )-strong spider in G with leaf vector ( u, a , . . . , a s − ) . (14) Claim 1.
Let ~a be a ( s − F ~a = ∅ . Let uw ∈ H ~a , where u ∈ U ~a and w ∈ W ~a . Then the number of members of F ′ containing uw is at least cδ ( Kδ ) k − and at most[ f ( k, L )] s − · ( Kδ ) k − . The number of members of F ′ containing w is at least cδ ( Kδ ) k − . Proof of Clam 1.
By definition, there is a member F ∈ F ~a such that w ( F ) = w and u ( F ) = u .Let F ∗ = F | ~a . Then F ∗ ∈ ∂ ( F ′ ). Since | F ∗ | = ( s − k + 2, by (13), there are at least cδ ( Kδ ) k − members of F ′ that contain F ∗ and hence contain uw . To upper bound the number of members of F ′ that contain uw , note that there are at most [ f ( k, L )] s − ways to pick the paths from a i to w for iang, Qiu: On Tur´an numbers of bipartite subdivisions i ∈ [ s −
1] and at most ( Kδ ) k − ways to grow such a member past u . Now, let S be obtained from F ∗ by deleting u . Then S ∈ ∂ ( F ′ ). Since | S | = ( s − k + 1, by (13), there are at least cδ ( Kδ ) k − members of F ′ that contain S and hence contain w . Claim 2.
For each ( s − ~a for which F ~a = ∅ , we have |F ~a | ≥ e ( H ~a ) · cδ ( Kδ ) k − and e ( H ~a ) ≥ c ′ δ · | W ~a | , where c ′ = cK/ [ f ( k, L )] s − . Proof of Claim 2.
By Claim 1, for each wu ∈ H ~a there are at least cδ ( Kδ ) k − members F of F ′ that contain wu . Since different wu ’s clearly give rise to different F ’s, the first part of the claimfollows.Now, let w ∈ W ~a . By Claim 1, there are at least cδ ( Kδ ) k − members of F ′ that contain w . Eachsuch member contains wu for some edge wu ∈ H ~a . On the other hand, for each such fixed wu , byClaim 1, there are at most [ f ( k, L ] s − ( Kδ ) k − members of F ′ that contain it. This implies that d H ~a ( w ) ≥ cδ ( Kδ ) k − [ f ( k, L )] s − · ( Kδ ) k − = c ′ δ. So e ( H ~a ) ≥ c ′ δ · | W ~a | .For any ( s − ~a = ( a , . . . , a s − ), let U + ~a = { u ∈ U ~a : d H ~a ( u ) ≥ kt } and U − ~a := { u ∈ U ~a : d H ~a ( u ) < kt. } . Let F + ~a = { F ∈ F ~a : u ( F ) ∈ U + ~a } , and F − ~a = { F ∈ F ~a : u ( F ) ∈ U − ~a } . Claim 3.
For every ( s − ~a we have e ( H ~a [ U + ~a , W ~a ]) ≤ kt | W ~a | . Proof of Claim 3.
Let ~a be given. For convenience, let U + = U + ~a and W = W ~a . Suppose that e ( H ~a [ U + , W ]) > kt | W | . Then this, together with the definition of U + ~a , implies that the averagedegree of H ~a [ U + , W ] is at least 2 kt . By a well-known fact, H ~a [ U + , W ] contains a subgraph H ′ withminimum degree at least kt . In H ′ , we can greedily build a t -legged spider T of height k − U . Let x be its center and u , . . . , u t be its leaves. By (14), ( u i , a , . . . , a s − ) is(1 , k, . . . , k )-strong for every i ∈ [ t ]. Thus using strong-ness one can greedily find t internally disjointbalanced spiders of height k with leaf vector ( x, a , . . . , a s − ). The union of these t spiders forms acopy of K ks,t , contradicting G being K ks,t -free.By Claims 1 and 3, we have |F + ~a | ≤ e ( H ~a [ U + ~a , W ~a ]) · [ f ( k, L )] s − ( Kδ ) k − ≤ [2 kt [ f ( k, L )] s − K k − ] · | W ~a | · δ k − . (15)On the other hand, by Claims 2 we have |F ~a | ≥ e ( H ~a ) · cδ ( Kδ ) k − ≥ c ′ δ | W ~a | · cδ ( Kδ ) k − = c ′ cK k − · | W ~a | · δ k . As δ ≥ Cn k − sk and n ≥ n is sufficiently large, this, together with (15) yields that |F + ~a | ≤ |F ~a | . iang, Qiu: On Tur´an numbers of bipartite subdivisions |F − ~a | = |F ~a | − |F + ~a | ≥ |F ~a | . Since F ′ = ∪ ~a F ~a , we have that P ~a |F ~a | ≥ |F ′ | ≥ γnδ sk . Itfollows that X ~a |F − ~a | ≥ X ~a |F ~a | ≥ γnδ ks ≥ γC sk n s . By averaging, there exists an ( s − ~a such that |F − ~a | ≥ C n , for some constant C that canbe made arbitrarily large by taking C to be sufficiently large. By averaging again, there exists some z such that the number of spiders in F − ~a with leaf vector ( ~a, z ) is at least C . Fix such a vertex z and let F ~a,z = { F ∈ F − ~a : F has leaf vector ( ~a, z ) } . Let W ~a,z = { w ( F ) : F ∈ F ~a,z } . Note that for each w ∈ W ~a,z , since members of F ~a,z by requirements contain no critical paths oflength at most k and hence no heavy paths of length at most k , the number of these members thathave w as the center and ( ~a, z ) as leaf vector is at most [ f ( k, L )] s . Hence | W ~a,z | ≥ |F ~a,z | [ f ( k, L )] s ≥ C [ f ( k, L )] s . By choosing C to be sufficiently large (which makes C sufficiently large) we can ensure | W ~a,z | ≥ [( sk ) k · f ( k, L ) · k ] k . Claim 4.
Some member of F ~a,z contains a ( j, k, . . . , k )-strong sub-spider for some 2 ≤ j ≤ k . Proof of Claim 4.
For each F ∈ F ~a,z , let P F denote the z, w ( F )-path in F . For each w ∈ W ~a,z ,by the definition of W ~a,z there exists some F ∈ F ~a,z such that w ( F ) = w . Fix such an F and let P w = P F . Let C = { P w : w ∈ W ~a,z } . Let m = ( sk ) k · f ( k, L ). Since | W ~a,z | ≥ ( mk ) k , by Lemma 3.11, for some j ∈ [ k ] there exist a vertex x and m vertices w , . . . , w m ∈ W ~a,z such that J := S i ∈ [ m ] P w i [ x, w i ] is a spider with center x andheight j . If j = 1, then J is a star of size at least m = ( sk ) sk · f ( k, L ) ≫ kt in H ~a [ U − ~a , W ~a ] with thecenter x ∈ U − ~a , contradicting the definition of U − ~a . Hence j ≥ x, ~a ) is ( j, k, . . . , k )-strong. As P w ∈ C , by the definition of C , thereexists some F ∈ F ~a,z such that P w = P F . In particular, w ( F ) = w . Let F ′ be the sub-spiderobtained from F by replacing P F with P F [ x, w ]. Then F ′ has leaf vector ( x, ~a ) and length vector( j, k, . . . , k ). If one can prove that the tuple ( x, ~a ) is ( j, k, . . . , k )-strong, then by definition, F ′ is( j, k, . . . , k )-strong and thus F contains a ( j, k, . . . , k )-strong sub-spider, which would prove the claim.Next, we show that indeed ( x, ~a ) is ( j, k, . . . , k )-strong. Let q = ( sk ) k − j · f ( k, L ) . By the definition of ( j, k, . . . , k )-strong-ness, we need to show there exist q internally disjoint spiderswith leaf vector ( x, ~a ) and length vector ( j, k, . . . , k ). For each i ∈ [ q ], let u i be the vertex on P w i [ x, w i ] iang, Qiu: On Tur´an numbers of bipartite subdivisions w i , and let P i = P w i [ x, u i ] for short. Note that for each i ∈ [ q ], u i ∈ U − ~a ⊆ U ~a andhence in particular ( u i , ~a ) is (1 , k, . . . , k )-strong. We will greedily find q spiders T , . . . , T q with lengthvector (1 , k, . . . , k ), satisfying that every T i has leaf vector ( u i , ~a ) and that T ∪ P , . . . , T q ∪ P q are q internally disjoint spiders with leaf vector ( x, ~a ) and length vector ( j, k, . . . , k ). Since ( u , ~a ) is(1 , k, . . . , k )-strong, there are at least ( sk ) k − · f ( k, L ) = ( sk ) j − q internally disjoint spiders with leafvector ( u , ~a ) and length vector (1 , k, . . . , k ). As | V ( ∪ qi =1 P i ) \{ u }| = ( j − q < ( sk ) j − q , there existsone such spider T such that V ( T ) ∩ V ( ∪ qi =1 P i ) = { u } . In general, suppose that for some p ≤ q wehave found T , . . . , T p − such that for each i ∈ [ p − T i is a spider with leaf vector ( u i , ~a ) and lengthvector (1 , k, . . . , k ) and V ( T i ) ∩ V ( ∪ qi =1 P i ) = { u i } and that V ( T i ) \ { a , . . . , a s } are disjoint over all i ∈ [ p − u p , ~a ) is (1 , k, . . . , k )-strong and | V ( ∪ qi =1 P i ) \ { u p }| = ( j − q , there are at least( sk ) j − q − ( j − q ≥ ( sk − q internally disjoint spiders T p with leaf vector ( u p , ~a ) and length vector(1 , k, . . . , k ), such that V ( T p ) ∩ V ( ∪ qi =1 P i ) = { u p } . Since the size of X := ∪ p − i =1 ( V ( T i ) \ { a , . . . , a s } )is ( sk − s − k + 2)( p − ≤ ( sk − s ) q , among these spiders there are at least( sk − q − ( sk − s ) q = ( s − q ≥ q spiders T p such that [ V ( T p ) \ { a , . . . , a s } ] ∩ X = ∅ . Hence, we can continue the process until wefind T , . . . , T q such that for each i ∈ [ q ] T i is a spider with leaf vector ( u i , ~a ) and length vector(1 , k, . . . , k ) and V ( T i ) ∩ V ( ∪ qi =1 P i ) = { u i } and that V ( T i ) \ { a , . . . , a s } are disjoint over all i ∈ [ q ].Now T ∪ P , . . . , T q ∪ P q are q internally disjoint spiders with leaf vector ( x, ~a ) and length vector( j, k, . . . , k ). The proof of Claim 4 is completed.By Claim 4, some member of F ~a,z contains a ( j, k, . . . , k )-strong sub-spider for some 2 ≤ j ≤ k ,which contradicts our assumption that no member of F contains any ( ℓ , . . . , ℓ s )-strong sub-spidersfor any ( ℓ , . . . , ℓ s ) = (1 , k, . . . , k ). This contradiction completes our proof of the lemma. The main idea of the proof of Theorem 1.2 is roughly as follows. In an almost regular graph withminimum degree δ ≥ Ω( n ) there are Ω( nδ ks ) ≥ Ω( n s ) balanced s -legged spiders of height k , that is,copies of K k ,s . Using the lemmas in the previous subsection as well as some new ones specific to the k = 3 , k or any strong sub-spiders. Using the pigeonhole principle, we can find an s -tuple that is the leafvector of a large number of K k ,s that do not contain strong sub-spiders or critical paths of length atmost k . This allows us to find at least t copies that are internally disjoint, whose union then forcesa copy of K ks,t . Lemma 3.13
Let k, s, t, L be positive integers. Let ℓ be an integer satisfying s ≤ ℓ ≤ sk . Let F bea family of spiders in a graph G that contain no critical path of length at most k and have the sameleaf vector ( v , . . . , v s ) and length vector ( ℓ , . . . , ℓ s ) . If |F | ≥ [( sk ) sk · f ( k, L ) ] ℓ then there exists amember of F that contains a strong sub-spider.Proof. We prove it by induction on ℓ . The case of ℓ = s is trivial. Assume ℓ > s and assume thatclaim holds for smaller ℓ values. Now pick a maximal family M of internally disjoint spiders in F . iang, Qiu: On Tur´an numbers of bipartite subdivisions |M| ≥ ( sk ) sk − ℓ · f ( k, L ), then any spider in M is strong (by Definition 3.4) and we are done. Sowe may assume |M| < ( sk ) sk − ℓ · f ( k, L ). Let U be the set of internal vertices of spiders in M . Then | U | ≤ sk · |M| < ( sk ) sk − ℓ +1 · f ( k, L ). By maximality of M , any spider in F contains a vertex in U .So by averaging, there exists u ∈ U such that the size of the family F which consists of all spidersin F that contain u is at least |F | ≥ |F || U | ≥ |F | ( sk ) sk − ℓ +1 · f ( k, L ) . By averaging again, there is a sub-family F ⊆ F of size |F | ≥ |F | ℓ − s + 1 ≥ |F | ( sk ) sk − ℓ +2 · f ( k, L ) . such that u plays the same role in members of F . Since any member of F contains no criticalpath of length at most k and hence no heavy paths of length at most k , there are no more than Q sj =1 f ( ℓ i , L ) ≤ [ f ( k, L )] s members of F that contain u as their center. It is easy to check thatby our assumption on |F | that |F | > [ f ( k, L )] s . So u cannot be the center of the spiders in F .Without loss of generality, we assume that u is in the first leg of the spiders in F and further assumethat in every F ∈ F the distance between of u and the center of F is ℓ ′ < ℓ . Now each member of F contains a sub-spider with leaf vector ( u, v , . . . , v s ) and length vector ( ℓ ′ , ℓ , . . . , ℓ s ). Let J bethe family of sub-spiders with leaf vector ( u, v , . . . , v s ) and length vector ( ℓ ′ , ℓ , . . . , ℓ s ) contained insome member of F . Since the members of F contain no critical path of length at most k , for any J ∈ J there are no more than f ( ℓ − ℓ ′ , L ) ≤ f ( k, L ) members of F containing J . It follows that |J | ≥ |F | f ( k, L ) ≥ |F | ( sk ) sk · f ( k, L ) ≥ [( sk ) sk · f ( k, L ) ] ℓ − . Since ℓ ′ + ℓ + · · · + ℓ s ≤ ℓ −
1, and |J | ≥ [( sk ) sk · f ( k, L ) ] ℓ − , by the induction hypothesis, thereexists a member T of J that contains a strong sub-spider. Now any member of F that contain T also contains a strong sub-spider. This completes the proof.We also need the following lemma that holds only for k = 3 ,
4. For its proof, let us first recallthe definitions of heavy paths and critical paths, given in Section 3.1.
Lemma 3.14
Suppose that F is an ( ℓ , . . . , ℓ s ) -strong spider, where ≤ ℓ ≤ · · · ≤ ℓ s ≤ k . If F contains no critical path of length at most k , then ℓ + ℓ ≥ k + 1 . Moreover, if k ∈ { , } , theneither ℓ ≥ k or ( ℓ , . . . , ℓ s ) = (1 , k, . . . , k ) .Proof. Let ℓ = ℓ + · · · + ℓ s . By Definition 3.2, every p -heavy path contains a q -critical path for some q ≤ p . Since F contains no critical path of length at most k , it contains no heavy paths of length atmost k . Suppose to a contrary that ℓ + ℓ ≤ k . Let ( v , . . . , v s ) be the leaf vector of F . Since F is strong, by Definition 3.4, there are at least ( sk ) sk − ℓ · f ( k, L ) > f ( k, L ) internally disjoint spiderswith leaf vector ( v , . . . , v s ) and length vector ( ℓ , . . . , ℓ s ). In particular, in their union, there exist atleast f ( k, L ) ≥ f ( ℓ + ℓ , L ) internally disjoint paths of length ℓ + ℓ joining v and v . This meansthat the path P in F that joins v and v is ( ℓ + ℓ )-heavy, contradicting our earlier discussion. iang, Qiu: On Tur´an numbers of bipartite subdivisions k ∈ { , } and ℓ < k . Then we have that ℓ = 1. Since ℓ + ℓ j ≥ ℓ + ℓ ≥ k + 1and ℓ j ≤ k for every 2 ≤ j ≤ s , it follows that ℓ j = k . Thus ( ℓ , . . . , ℓ s ) = (1 , k, . . . , k ).For k ∈ { , } , we can now combine Corollary 3.9 and Lemma 3.12 to obtain the following. Corollary 3.15
Let k ∈ { , } , K ≥ and integers s, t ≥ be fixed. Then provided that L issufficiently large compared to s, t, k and K , for any ζ > there exist C, n > such that the followingholds. Suppose that G is an K ks,t -free K -almost-regular graph n ≥ n vertices with minimum degree δ ≥ Cn k − sk . Let F denote the family of s -legged spiders of k that contain a strong sub-spider butcontain no critical path of length at most k . Then |F | ≤ ζnδ sk . Now, we are finally ready to prove our main theorem.
Proof of Theorem 1.2:
First we set some constants. Fix integers s, t ≥ k ∈ { , } . Let K be obtained by Lemma 2.1 with ǫ = k − sk . Let c = c ( k, s ) such that c equals the cardinality ofthe automorphism group of the s -legged spider of height k . Choose L to be a large constant suchthat Lemma 3.3 and Corollary 3.15 are valid. We further require that L is large enough such that K sk f (1 ,L ) ≤ c k . Let C = [( sk ) sk · f ( k, L ) ] sk , that is, be the constant in Lemma 3.13 with ℓ = sk . Let C be a large constant such that Corollary 3.15 holds with ζ := c . We further require that C is largeenough such that cC sk ≥ C .By Lemma 2.1, it suffices to show the following statement. For sufficiently large n , if G is an n -vertex K -almost-regular graph with minimum degree δ ≥ Cn k − sk , then G contains a copy of K ks,t .We will prove this by contradiction. Suppose to the contrary that G is K ks,t -free. Let F be thefamily of all the s -legged spiders of height k in G . Then by a greedy process, it is easy to see that |F | ≥ cn sk − Y i =0 ( δ − i ) ≥ c nδ sk , (16)where the last inequality holds because δ ≥ Cn k − sk and n is sufficiently large. By Lemma 3.3,for every 2 ≤ ℓ ≤ k , the number of critical paths of length ℓ is at most f ( ℓ − ,L ) n ( Kδ ) ℓ . Since themaximum degree of G is at most Kδ , the number of members of F that contain a critical pathof length ℓ is at most f ( ℓ − ,L ) n ( Kδ ) ℓ · ( Kδ ) sk − ℓ = K sk f ( ℓ − ,L ) nδ sk ≤ K sk f (1 ,L ) nδ sk ≤ c k nδ sk , where theinequality holds by the choice of L . So the number of members of F that contain a critical path oflength at most k is no more than ( k − · c k nδ sk < c nδ sk . Let F ′ denote the family of members of F that contain no critical path of length at most k . It follows that |F ′ | ≥ |F | − c nδ sk ≥ (cid:16) c − c (cid:17) nδ sk ≥ c nδ sk , (17)where in the second inequality we used (16).Let F ′′ denote the family of spiders in F ′ that contain no strong sub-spider. By Corollary 3.15we have that |F ′ \ F ′′ | ≤ ζnδ sk = c nδ sk , iang, Qiu: On Tur´an numbers of bipartite subdivisions ζ . This, together with (17), gives us that |F ′′ | = |F ′ | − |F ′ \ F ′′ | ≥ c nδ sk − c nδ sk = c nδ sk . Since δ ≥ Cn k − sk , it follows that |F ′′ | ≥ c nδ ks ≥ cC ks n s ≥ C n s , where the last inequality holds because of the choice of C . By averaging, there exists a tuple( v , . . . , v s ) of distinct vertices such that the subfamily F of F ′′ that consist of all the members of F ′′ that have leaf vector ( v , . . . , v s ) has size at least |F | ≥ |F ′′ | n s ≥ C n s n s = C . Now F is a family of spiders that have the same leaf vector and contain no critical path of length atmost k . Since |F | ≥ C , by the definition of C given at the beginning of the proof and Lemma 3.13,there exists a member of F ⊆ F ′′ that contains a strong sub-spider. This contradicts our definitionof F ′′ and completes the proof. It is easy to derive from the discussions in Sections 3.1, 3.2, and 3.4 the following weakening ofConjecture 1.1.
Proposition 4.1
Let s, t, k ≥ be integers. Let K ≤ ks,t denote the family of graphs that can be beobtained from K s,t by replacing each edge uv with a path of length at most k between u and v so thatthe st replacing paths are internally disjoint. Then ex ( n, K ≤ ks,t ) = O ( n k − sk ) . This together with the general theorem of Bukh and Conlon [2] implies the following.
Corollary 4.2
Let s, t, k ≥ be integers. Then ex ( n, K ≤ ks,t ) = Θ( n k − sk ) . The authors thank Jozsef Balogh for his valuable comments on an earlier version of the paper, inparticular, for suggesting the question leading to Proposition 4.1.
References [1] B. Bukh, Random algebraic construction of extremal graphs,
Bull. Lond. Math. Soc. (2015), 939-945.[2] B. Bukh and D. Conlon, Rational exponents in extremal graph theory,
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On Tur´an numbers of bipartite subdivisions
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