The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments

The t ypical structure of orien ted graphs and digraphs with forbidden blo w-up of transitiv e tournamen ts Jianxi Liu ∗ Abstract W e study the t ypical structure of orien ted graphs and digraphs that do not con tain a blow-up T t r +1 of a transitiv e tournamen t. F or any integers r ≥ 2, t ≥ 1 and an y real a ∈ (3 / 2 , 2], we pro ve that almost all T t r +1 -free oriented graphs and almost all T t r +1 -free digraphs are r -partite. This extends the results of K¨ uhn, Osth us, T o wnsend and Zhao (2017) on forbidden transitiv e tournaments to their blow-ups, thereb y con- firming a generalised form of Cherlin’s conjecture. Our proof com bines the h yp ergraph con tainer metho d, a w eighted analogue of the Erd˝ os–Stone theorem for digraphs, and a stability analysis for near-extremal T t r +1 -free digraphs. The core of the pro of is the in terplay betw een the directed regularity lemma and an embedding lemma, whic h to- gether provide a rigorous bridge from macroscopic extremal conditions to microscopic concrete structures. 1 In tro duction Giv en a fixed (di)graph H , a (di)graph is called H -fr e e if it do es not contain H as a subgraph. Extremal graph theory in vestigates tw o central questions: (1) What is the maxim um n umber of edges in an H -free graph on n vertices? (2) What is the t ypical structure of an H -free graph on n vertices? F or undirected graphs, Erd˝ os, Kleitman and Rothschild [4] initiated the study of the second question by proving that almost all triangle-free graphs are bipartite, and asymptotically determined the num b er of K r +1 -free graphs. Kolaitis, Pr¨ omel and Rothschild [5] strengthened this by sho wing that for ev ery r ≥ 2, almost all K r +1 -free graphs are r -partite. F or directed graphs (digraphs) and oriented graphs, the situation is far more complex. A digraph consists of a set of vertices and a set of ordered pairs of distinct v ertices (no lo ops or multiple arcs in the same direction). An oriented graph is a digraph with at most one arc b et ween any t wo vertices, i.e., it is an orien tation of a simple undirected graph. The transitive tournament T k is the orientation of a complete graph K k that is transi- tiv e (the v ertices can b e linearly ordered so that all arcs go from smaller to larger v ertices). In his work on coun table homogeneous oriented graphs, Cherlin [3] noted that the striking results of [5] do not seem to extend directly to the directed case, and he made the follo wing conjectures. Conjecture 1.1 (Cherlin). (i) Almost all T 3 -free oriented graphs are tripartite. (ii) Almost all C 3 -free oriented graphs are acyclic, i.e. they are subgraphs of transitiv e tournaments. ∗ Sc ho ol of Mathematics and Statistics, Guangdong Universit y of F oreign Studies, Guangzhou, China. Email: jxliu@gdufs.edu.cn 1 K ¨ uhn, Osthus, T o wnsend and Zhao [6] (henceforth KOTZ) confirmed part (i) of this conjecture and, more generally , prov ed that for every k ≥ 2, almost all T k +1 -free orien ted graphs are k -partite; they also obtained the analogous result for digraphs. Note that this sho ws that in fact almost all T 3 -free oriented graphs are actually bipartite – a structure quite differen t from the extremal T 3 -free orien ted graph, which is the blow-up of a directed triangle. A natural generalisation is to forbid the blow-up of a transitive tournament. F or in tegers r, t ≥ 1, let T t r +1 denote the digraph obtained from T r +1 b y replacing eac h v ertex with an indep enden t set of size t and adding, for each arc ij of T r +1 , all p ossible arcs from the i -th part to the j -th part. Thus T t r +1 is an oriented graph (no 2-cycles) when t ≥ 1. Clearly T t r +1 con tains T r +1 as a subgraph (take one v ertex from each part), so an y T r +1 -free digraph is also T t r +1 -free, but the con verse is false. In this pap er w e determine the typical structure of T t r +1 -free oriented graphs and digraphs. Our main result shows that even forbidding these larger structures do es not c hange the asymptotic picture: almost all suc h graphs are r -partite. This provides a far-reac hing extension of the KOTZ theorem and confirms a generalised form of Cherlin’s conjecture. Theorem 1.1 (Main result – simplified form) . F or any inte gers r ≥ 2 and t ≥ 1 , almost al l T t r +1 -fr e e oriente d gr aphs ar e r -p artite, and almost al l T t r +1 -fr e e digr aphs ar e r -p artite. A more precise v ersion, in volving the w eight parameter a (which unifies the treat- men t of oriented graphs and digraphs) and giving exp onen tial bounds on the num ber of exceptions, reads as follo ws. Theorem 1.2 (Main result – precise form) . L et r ≥ 2 , t ≥ 1 b e inte gers and let a ∈ (3 / 2 , 2] . Then for every α > 0 ther e exists ε > 0 such that for al l sufficiently lar ge n the fol lowing hold. • A l l but at most f ( n, T t r +1 ) · 2 − εn 2 lab el le d T t r +1 -fr e e oriente d gr aphs on n vertic es c an b e turne d into an r -p artite oriente d gr aph by changing at most αn 2 ar cs. • A l l but at most f ∗ ( n, T t r +1 ) · 2 − εn 2 lab el le d T t r +1 -fr e e digr aphs on n vertic es c an b e turne d into an r -p artite digr aph by changing at most αn 2 ar cs. In p articular, f ( n, T t r +1 ) = T ( n, r )(1 + o (1)) and f ∗ ( n, T t r +1 ) = T ∗ ( n, r )(1 + o (1)) , wher e T ( n, r ) (r esp. T ∗ ( n, r ) ) is the numb er of lab el le d r -p artite oriente d gr aphs (r esp. digr aphs) on n vertic es. The pro of com bines three p o werful tools: the hypergraph container method (Theo- rem 2.3), a w eigh ted analogue of the Erd˝ os–Stone theorem for digraphs (Theorem 3.1), and a stabilit y analysis for near-extremal T t r +1 -free digraphs (Theorem 4.1). The core of the pro of is the interpla y b et ween the directed regularity lemma and an embedding lemma, whic h together provide a rigorous bridge from macroscopic extremal conditions to microscopic concrete structures. Pro of outline The pro of follo ws a three-step strategy . 1. Con tainer metho d. W e apply the hypergraph con tainer theorem (Theorem 2.3) to obtain a small family C of digraphs (called con tainers) suc h that every T t r +1 -free digraph is con tained in some mem b er of C , and eac h container contains few copies of T t r +1 and has weigh ted size close to the extremal v alue. 2 2. Cleaning containers. Using the remo v al lemma (Lemma 2.4), w e delete a few arcs from eac h container to obtain a genuinely T t r +1 -free digraph whose w eighted size is still near the extremal v alue. 3. Stabilit y argument. The stabilit y theorem (Theorem 4.1) then implies that each suc h cleaned digraph is close to the complete r -partite digraph DT r ( n ). Conse- quen tly , the original containers, and hence all but a tin y fraction of T t r +1 -free di- graphs, are close to DT r ( n ). A more precise inductive coun ting argumen t (see Section 5.2) upgrades this approximate structural result to the exact statemen t that almost all T t r +1 -free (orien ted) digraphs are r -partite. The pap er is organised as follo ws. Section 2 in tro duces notation and k ey to ols. Section 3 pro ves a w eigh ted extremal theorem for T t r +1 (Theorem 3.1), whic h is a directed analogue of the Erd˝ os–Stone theorem. Section 4 establishes a stabilit y result for T t r +1 -free digraphs (Theorem 4.1). Section 5 combines the con tainer theorem with stability to pro v e the main result. Section 6 concludes with remarks and op en problems. 2 Preliminaries 2.1 Notation F or a digraph G = ( V , E ), let f 1 ( G ) b e the num b er of unordered pairs { u, v } such that exactly one of uv and v u b elongs to E , and let f 2 ( G ) b e the num b er of unordered pairs with b oth uv and v u present. F or a real num b er a ≥ 1, define the weighte d size e a ( G ) := a · f 2 ( G ) + f 1 ( G ) . This unifies the treatmen t of oriented graphs ( a = log 3, since eac h 2-cycle contributes 3 w ays to orien t) and digraphs ( a = 2, total num ber of arcs). Let ex a ( n, H ) denote the maxim um e a ( G ) ov er all H -free digraphs on n vertices. Let T u r ( n ) b e the r -partite T ur´ an graph on n v ertices (parts as equal as p ossible), and let t r ( n ) = e (T u r ( n )). Denote b y DT r ( n ) the digraph obtained from T u r ( n ) by replacing each undirected edge with a pair of opp osite arcs. Clearly DT r ( n ) is r -partite and T t r +1 -free, so ex a ( n, T t r +1 ) ≥ a t r ( n ). W e sa y that almost al l graphs in a family F ha ve prop ert y P if lim n →∞ |{ G ∈ F n : G has prop erty P }| |F n | = 1 . 2.2 Directed regularit y and em b edding lemmas W e will need the directed version of Szemer´ edi’s regularit y lemma and an embedding lemma for blow-ups. The following form ulation is from [1]. Lemma 2.1 (Directed regularity lemma) . F or any ε ∈ (0 , 1) and inte gers M ′ , M ′′ , ther e exist M and n 0 such that for any digr aph G on n ≥ n 0 vertic es, any initial p artition U 0 , U 1 , . . . , U M ′′ , and any d ∈ [0 , 1] , ther e exists a p artition V 0 , V 1 , . . . , V k of V ( G ) and a sp anning sub digr aph G ′ ⊆ G with: • M ′ ≤ k ≤ M , | V 0 | ≤ εn , | V 1 | = · · · = | V k | = ℓ ; • G ′ [ V i ] is empty for al l i ≥ 1 ; 3 • for 1 ≤ i  = j ≤ k , the bip artite digr aph ( V i , V j ) G ′ is either ε -r e gular with density at le ast d , or has density 0 ; • every vertex x satisfies d + G ′ ( x ) > d + G ( x ) − ( d + ε ) n and similarly for in-de gr e e. The r e duc e d digr aph R has vertex set { V 1 , . . . , V k } and an arc ij whenever ( V i , V j ) G ′ is ε -regular with density ≥ d . Lemma 2.2 (Embedding lemma) . F or any d ∈ (0 , 1) and maximum de gr e e ∆ ≥ 1 , ther e exists ε 0 > 0 such that the fol lowing holds. L et G b e a digr aph, R the r e duc e d digr aph obtaine d fr om an ε -r e gular p artition with ε ≤ ε 0 , cluster size ℓ , and density p ar ameter d . If H is a digr aph with ∆( H ) ≤ ∆ and H ⊆ R s (the blow-up of R wher e e ach vertex is r eplac e d by s indep endent vertic es), and ℓ ≥ s/ε 0 , then H ⊆ G . (se e e.g. [6, L emma 4.2]) 2.3 Hyp ergraph con tainers The following con tainer theorem for general digraphs under a natural sparsit y condition w as prov ed b y Liu [8]. It extends the earlier result of K ¨ uhn, Osth us, T o wnsend and Zhao [6] from oriented graphs to digraphs that satisfy a density condition. • Condition A (sparsity). F or every subgraph H ′ ⊆ H with e ( H ′ ) > 1, e ( H ′ ) v ( H ′ ) ≤ a 2 , where a is the same parameter as in the definition of e a . When a = 2, Condition A requires e/v ≤ 1, which means H has no 2-cycles; th us the theorem reduces to the orien ted case. F or a > 2, Condition A allows digraphs with a controlled density of 2-cycles. F or example, taking a = 4, we ma y hav e e/v ≤ 2, so H can con tain 2-cycles as long as its ov erall densit y is b ounded. This pro vides a genuine extension b eyond oriented graphs. Theorem 2.3 (Liu [8]) . L et H b e a digr aph satisfying Condition A, with h = v ( H ) , e ( H ) ≥ 2 , and let a ≥ 1 . F or every ε > 0 ther e exists c > 0 such that for al l sufficiently lar ge N ther e exists a c ol le ction C of digr aphs on [ N ] with the fol lowing pr op erties. (a) Every H -fr e e digr aph I on [ N ] is c ontaine d in some G ∈ C . (b) Every G ∈ C c ontains at most εN h c opies of H , and e a ( G ) ≤ ex a ( N , H ) + εN 2 . (c) log |C | ≤ cN 2 − 1 /m ( H ) log N , wher e m ( H ) = max H ′ ⊆ H, e ( H ′ ) > 1 e ( H ′ ) − 1 v ( H ′ ) − 2 . 2.4 Remo v al lemma W e also need the directed remo v al lemma of Alon and Shapira [1]. Lemma 2.4 (Remov al lemma) . F or any fixe d digr aph H on h vertic es and any γ > 0 , ther e exists ε ′ > 0 such that for al l lar ge n , any digr aph G on n vertic es c ontaining at most ε ′ n h c opies of H c an b e made H -fr e e by deleting at most γ n 2 ar cs. 4 3 W eigh ted extremal theorem for T t r +1 In this section w e prov e a weigh ted Erd˝ os–Stone type theorem for the blow-up T t r +1 . Theorem 3.1. F or any inte gers r , t ≥ 1 , r e al a ∈ (3 / 2 , 2] , and γ > 0 , ther e exists n 0 such that for al l n ≥ n 0 , every digr aph G on n vertic es with e a ( G ) ≥ a t r ( n ) + γ n 2 c ontains T t r +1 as a sub digr aph. Pr o of. W e follo w the strategy of the KOTZ pro of for T r +1 . Set d = γ / 4, ∆ = ∆( T t r +1 ), and let ε 0 b e given by Lemma 2.2 for these parameters. Cho ose ε small enough so that ε ≤ ε 0 and δ := ( a − 1) d − ε − aε 2 / 2 − aε > 0. Let s = t ( r + 1). Apply Lemma 2.1 to G with parameters ε, d to obtain a partition V 0 , V 1 , . . . , V k and a pure digraph G ′ ⊆ G with clusters of size ℓ , where ℓ ≥ (1 − ε ) n/k ≥ s/ε 0 for large n . Let R b e the reduced digraph on k v ertices. F rom the regularity lemma w e hav e e a ( G ) ≤ e a ( G ′ ) + ( d + ε ) n 2 ≤ aεn 2 + e ∗ a ( R ) ℓ 2 + ( d + ε ) n 2 , where e ∗ a ( R ) = P ij ∈ E ( R ) ( ad 2 ij + d 1 ij ) and d 2 ij , d 1 ij are the densities of 2-cycles and 1-cycles in the pair ( V i , V j ) G ′ . Using e a ( G ) ≥ at r ( n ) + γ n 2 and t r ( n ) =  1 − 1 r  n 2 2 + O ( n ), we obtain after elementary manipulation e ∗ a ( R ) ≥ a  1 − 1 r  k 2 2 + δ k 2 > a t r ( k ) . No w R is T r +1 -free: otherwise, by the embedding lemma, G ′ w ould contain T t r +1 . The inequality e ∗ a ( R ) > a t r ( k ) forces R to contain T r +1 (b y the extremal result of Bro wn–Harary [2] for a = 2 and its weigh ted version for a ∈ (3 / 2 , 2] pro ved in [6]). Hence R ⊇ T r +1 , and then R t con tains T t r +1 . Applying the em b edding lemma with H = T t r +1 giv es T t r +1 ⊆ G ′ ⊆ G . As an immediate corollary w e obtain the asymptotics of the w eighted T ur´ an num b er. Corollary 3.2. F or r, t ≥ 1 and a ∈ (3 / 2 , 2] , we have ex a ( n, T t r +1 ) = a t r ( n ) + o ( n 2 ) . Mor e over, any n -vertex T t r +1 -fr e e digr aph G with e a ( G ) = ex a ( n, T t r +1 ) differs fr om DT r ( n ) by o ( n 2 ) ar cs. 4 Stabilit y for T t r +1 -free digraphs In this section we pro ve a stabilit y result for T t r +1 -free digraphs. It states that any suc h digraph whose weigh ted size is close to the extremal v alue a t r ( n ) m ust b e close to the complete r -partite digraph DT r ( n ). Theorem 4.1 (Stability theorem) . L et r ≥ 2 , t ≥ 1 and a ∈ (3 / 2 , 2] . F or every β > 0 ther e exist γ > 0 and n 0 such that for al l n ≥ n 0 , if G is an n -vertex T t r +1 -fr e e digr aph satisfying e a ( G ) ≥ a t r ( n ) − γ n 2 , then G c an b e turne d into DT r ( n ) by changing at most β n 2 ar cs. The pro of follows a well-established pattern: we apply the directed regularit y lemma to obtain a reduced digraph R , transfer the w eighted extremal condition to R , use a w eighted v ersion of the stability theorem for T r +1 (whic h forces R to b e close to DT r ( m )), and finally lift this structure back to G . The main difficulty is to con trol the w eights, whic h is handled by a “w eighted stability lemma” (Lemma 4.2) for T r +1 -free digraphs. W e b egin with the statemen t of this auxiliary result. 5 4.1 A w eigh ted stabilit y lemma for T r +1 F or a digraph R on m vertices w e denote b y e ∗ a ( R ) the quan tity e ∗ a ( R ) = X ij ∈ E ( R ) w ij , where w ij ∈ [1 , a ] is a weigh t assigned to the arc ij . In our application w ij will b e the a verage of a times the densit y of 2-cycles plus the density of 1-cycles ov er a regular pair. The follo wing lemma is the analogue of the ordinary stabilit y theorem for T r +1 but with w eights. Lemma 4.2 (W eigh ted stabilit y lemma) . F or every r ≥ 2 , a ∈ (3 / 2 , 2] and η > 0 ther e exist δ > 0 and m 0 such that for al l m ≥ m 0 the fol lowing holds. L et R b e an m -vertex T r +1 -fr e e digr aph and assign to e ach ar c ij ∈ E ( R ) a weight w ij ∈ [1 , a ] in such a way that e ∗ a ( R ) ≥ a t r ( m ) − δ m 2 . Then ther e exists a p artition U 1 , . . . , U r of the vertex set of R with the fol lowing pr op erties, up to at most η m 2 exc eptions: (i) if i, j b elong to the same class U p , then R c ontains no ar c b etwe en i and j ; (ii) if i ∈ U p , j ∈ U q with p  = q , then b oth ar cs ij and j i ar e pr esent in R and their weights satisfy w ij , w j i ≥ a − η . 4.1.1 Pro of of Lemma 4.2 W e argue by contradiction. Supp ose the statement is false. Then there exist constants η 0 > 0 and a sequence of coun terexamples: for every k w e can find m k → ∞ , a T r +1 -free digraph R k on m k v ertices, and w eights w ( k ) ij ∈ [1 , a ] such that e ∗ a ( R k ) ≥ a t r ( m k ) − 1 k m 2 k (i.e. δ k = 1 /k → 0) , but R k do es **not** admit a partition U 1 , . . . , U r with the required prop erties for η 0 (i.e., for ev ery r -partition there are at least η 0 m 2 k violating pairs). W e will derive a con tradiction. Step 1: Regularisation of R k . Apply the directed regularity lemma (Lemma 2.1) to R k with parameters ε and d that will b e chosen later (v ery small compared to η 0 ). W e obtain a partition V ( k ) 0 , V ( k ) 1 , . . . , V ( k ) p k of V ( R k ) and a pure digraph R ′ k ⊆ R k with the usual prop erties: | V ( k ) 0 | ≤ εm k , | V ( k ) 1 | = · · · = | V ( k ) p k | = ℓ k , p k ≥ M ′ , and for every i  = j , the pair ( V ( k ) i , V ( k ) j ) R ′ k is either ε -regular with densit y at least d or has density 0. Denote p = p k and let e R k b e the reduced digraph on { 1 , . . . , p } (arcs corresp ond to regular pairs of density ≥ d ). Step 2: T ransferring w eights to e R k . F or eac h ordered pair ( i, j ) that is an arc of e R k , define e w ( k ) ij = 1 | V ( k ) i || V ( k ) j | X u ∈ V ( k ) i ,v ∈ V ( k ) j w ( k ) uv , where w ( k ) uv is the w eight of the arc uv in R k (if uv / ∈ E ( R k ) we treat w ( k ) uv = 0). Since all w ( k ) uv ∈ [1 , a ], we ha ve e w ( k ) ij ∈ [1 , a ] as w ell. Moreov er, b y the prop erties of the regularit y 6 lemma, the contribution of all arcs not co vered b y regular pairs is negligible; a standard calculation (see e.g. [ ? , Lemma 9.2]) giv es e ∗ a ( e R k ) ≥ a t r ( p ) − δ ′ k p 2 , where δ ′ k → 0 as k → ∞ (pro vided ε and d are c hosen small enough relative to δ k ). In particular, for large k w e hav e δ ′ k ≤ δ 0 for any prescrib ed δ 0 > 0. Step 3: Structure of e R k . Because R k is T r +1 -free, the reduced digraph e R k is also T r +1 -free (otherwise an embedding argumen t would pro duce a T r +1 in R k ). Moreov er, the n umber p of clusters is b ounded by some absolute constant M depending only on ε and the initial parameters; indeed, the regularit y lemma guaran tees p ≤ M . Th us p is b ounded indep enden tly of k . F or each fixed p , there are only finitely many T r +1 -free digraphs on p v ertices. The w eighted extremal num b er ex a ( p, T r +1 ) equals at r ( p ) and is uniquely attained by DT r ( p ) (by Lemma 4.1 in [6], whic h holds for all p ). Consequently , if δ ′ k is smaller than the minimum p ositive difference b etw een at r ( p ) and the weigh ted size of an y non-extremal T r +1 -free digraph on p v ertices, then e ∗ a ( e R k ) ≥ at r ( p ) − δ ′ k p 2 forces e R k ∼ = DT r ( p ). Since δ ′ k → 0, for sufficien tly large k this condition is satisfied, and we obtain e R k ∼ = DT r ( p ). Let the parts of this DT r ( p ) be W 1 , . . . , W r (eac h W i is a set of cluster indices). Step 4: Lifting to R k . Now define a partition U 1 , . . . , U r of V ( R k ) by putting ev ery v ertex b elonging to a cluster with index in W i in to U i , and distributing the v ertices of the exceptional set V ( k ) 0 arbitrarily (e.g., equally). W e claim that this partition satisfies the required prop erties with at most η 0 m 2 k exceptions, contradicting the c hoice of R k . Consider an y tw o v ertices x, y not b oth in V ( k ) 0 . If they lie in differen t clusters V i , V j with i ∈ W p , j ∈ W q and p  = q , then b ecause e R k has b oth arcs ij and j i , the pair ( V i , V j ) is ε -regular with density at least d . Moreov er, from e ∗ a ( e R k ) = at r ( p ) and the uniqueness of the extremal digraph, w e actually hav e e w ij = a for every arc ij of e R k . By definition of e w ij , this implies that for ev ery u ∈ V i , v ∈ V j , the original weigh t w uv = a ; hence the arc uv is presen t and has weigh t a , i.e., it is a 2-cycle with densit y 1 (since ad 2 + d 1 = a and a > 1, the only p ossibility is d 2 = 1, d 1 = 0). Consequently , the pair ( V i , V j ) is a complete bidirectional pair (every p ossible arc in b oth directions is pres en t) and contributes aℓ 2 to e a ( R ′ k ); in R k the same holds up to the degree loss b ound, which is negligible. No w we b ound the num b er of violating pairs: • P airs inside V ( k ) 0 : at most | V ( k ) 0 | 2 ≤ ε 2 m 2 k . • P airs with one v ertex in V ( k ) 0 and the other outside: at most 2 εm 2 k . • P airs inside the same cluster V i : in R ′ k there are no such arcs; in R k the total n um b er of arcs inside clusters is b ounded by the degree loss, at most ( d + ε ) m 2 k / 2. • P airs coming from clusters that are in the same part W p but differen t clusters: in e R k there are no arcs b etw een them; in R k the arcs b et ween such clusters are again b ounded by the degree loss, at most ( d + ε ) m 2 k . Th us the total n umber of arcs violating the ideal structure is at most ( ε 2 + 2 ε + ( d + ε ) / 2 + ( d + ε )) m 2 k , which can b e made smaller than η 0 m 2 k b y choosing ε, d sufficiently small. This con tradicts the assumption that R k w as a coun terexample, completing the pro of of Lemma 4.2. 7 4.2 Regularit y setup for Theorem 4.1 W e no w start the pro of of Theorem 4.1. Fix r, t and a as in the theorem, and let β > 0 b e given. W e will choose a c hain of constan ts 1 n 0 ≪ ε ≪ d ≪ η ≪ β , 1 r , where η will b e the parameter app earing in Lemma 4.2. The exact dep endencies will b ecome clear during the pro of. Set s = t ( r + 1) and let ∆ = ∆( T t r +1 ). By the embedding lemma (Lemma 2.2) there exists ε 0 = ε 0 ( d, ∆) suc h that if ε ≤ ε 0 and the cluster size ℓ ≥ s/ε 0 , then any blo w-up of a sub digraph of the reduced digraph can b e embedded. W e c ho ose ε ≤ min { ε 0 , d } . Apply the directed regularity lemma (Lemma 2.1) to G with paramete rs ε, d . W e obtain a partition V 0 , V 1 , . . . , V k of V ( G ) and a pure digraph G ′ ⊆ G with the usual prop erties: | V 0 | ≤ εn , | V 1 | = · · · = | V k | = ℓ , k ≥ M ′ (some absolute constant) and ev ery pair ( V i , V j ) G ′ is either ε -regular with densit y at least d or has densit y 0. Denote m = k and set R to b e the reduced digraph on v ertex set { 1 , . . . , m } where ij is an arc iff ( V i , V j ) G ′ is ε -regular with density ≥ d . F or eac h such regular pair we define d 2 ij = # { ordered pairs ( u, v ) ∈ V i × V j with b oth uv , v u ∈ G ′ } | V i || V j | , d 1 ij = # { ordered pairs ( u, v ) ∈ V i × V j with exactly one of uv , v u ∈ G ′ } | V i || V j | , so that d 1 ij + d 2 ij ≥ d . The weigh ted contribution of this pair to e a ( G ′ ) is ( ad 2 ij + d 1 ij ) ℓ 2 . W e define a weigh t on the arc ij of R b y w ij = ad 2 ij + d 1 ij . Clearly w ij ∈ [ d, a ]. Moreov er, for an ordered pair ( i, j ) that is not an arc of R we set w ij = 0 (it will pla y no role). The w eighted size of R (with these w eights) is e ∗ a ( R ) = X ij ∈ E ( R ) w ij . 4.3 F rom G to R W e no w relate e a ( G ) to e ∗ a ( R ). F rom the regularity lemma w e hav e e a ( G ) ≤ e a ( G ′ ) + ( d + ε ) n 2 . Inside G ′ the only p ossible arcs are b et ween different clusters, and those that b elong to non-regular or low-densit y pairs contribute nothing to e a ( G ′ ) by definition. Hence e a ( G ′ ) ≤ a | V 0 | n + X ij ∈ E ( R ) w ij ℓ 2 ≤ aεn 2 + e ∗ a ( R ) ℓ 2 . Com bining these inequalities yields e a ( G ) ≤ e ∗ a ( R ) ℓ 2 + ( aε + d + ε ) n 2 . (1) 8 On the other hand the hypothesis of Theorem 4.1 gives e a ( G ) ≥ a t r ( n ) − γ n 2 . Using the well-kno wn estimate t r ( n ) =  1 − 1 r  n 2 2 + O ( n ) and the fact that ℓ ≥ (1 − ε ) n/m , w e obtain from (1) e ∗ a ( R ) (1 − ε ) 2 n 2 m 2 ≥ a  1 − 1 r  n 2 2 − γ n 2 − ( aε + d + ε ) n 2 − O ( n ) . After multiplying by m 2 /n 2 and setting δ := γ + aε + d + ε + o (1) , w e get e ∗ a ( R ) ≥ a  1 − 1 r  m 2 2 − δ m 2 . (2) 4.4 R is T r +1 -free Supp ose for a contradiction that R contains a copy of T r +1 . Then, by the embedding lemma (Le mma 2.2) with s = t ( r + 1), we can em b ed T t r +1 in to G ′ , because each regular pair has density at least d and the cluster size ℓ is at least s/ε 0 (since n is large enough). This would contradict the fact that G (and hence G ′ ) is T t r +1 -free. Therefore R is T r +1 -free. 4.5 Applying the weigh ted stabilit y lemma to R No w we apply Lemma 4.2 to R with the w eights w ij defined abov e. By (2), if we choose γ (and consequently δ ) sufficiently small, the hypothesis of the lemma is satisfied with η (whic h w e ha v e not y et fixed; w e will choose it later). Consequently there exists a partition U 1 , . . . , U r of [ m ] such that, with at most η m 2 exceptional pairs, we ha ve: • no arcs inside the same U p ; • for p  = q and i ∈ U p , j ∈ U q , b oth arcs ij and j i belong to E ( R ) and w ij , w j i ≥ a − η . 4.6 Lifting the partition to G Using this partition of the clusters w e now define a partition X 1 , . . . , X r of V ( G ): put all vertices of V i in to X p exactly when i ∈ U p . The exceptional v ertices in V 0 can b e distributed arbitrarily (e.g. equally among the X p ). W e will sho w that G differs from DT r ( n ) by at most β n 2 arcs; here DT r ( n ) means the complete r -partite digraph in which ev ery cross pair contains b oth directions and there are no arcs inside parts. W e need to b ound the n umber of arcs that violate this ideal structure. They can be classified as follows. 1. Arcs inciden t to V 0 . Since | V 0 | ≤ εn , there are at most 2 εn 2 suc h arcs. 2. Arcs coming from pairs that are not regular or hav e lo w densit y . In G ′ these pairs contribute nothing; in the original G they can ha ve at most 2( d + ε ) n 2 arcs, b ecause every vertex loses at most ( d + ε ) n neigh b ours in each direction when passing from G to G ′ . 9 3. Arcs corresp onding to exceptional cluster pairs. By the conclusion of Lemma 4.2, there are at most η m 2 unordered pairs { i, j } that violate either the “no arc inside a part” condition or the “full bidirectional arcs with large weigh t” condition. Each suc h pair inv olv es at most 2 ℓ 2 arcs (b oth directions). Hence the total num ber of arcs in this category is at most 2 η m 2 ℓ 2 ≈ 2 η n 2 . 4. Arcs in “go o d” pairs that are not y et complete bidirectional. Consider a pair ( i, j ) with i ∈ U p , j ∈ U q , p  = q , for which b oth arcs exist and w ij , w j i ≥ a − η . What do es w ij ≥ a − η imply ab out the actual densities d 2 ij , d 1 ij ? Since w ij = ad 2 ij + d 1 ij and d 1 ij = d ij − d 2 ij ≤ 1 − d 2 ij , we hav e ad 2 ij + 1 − d 2 ij ≥ a − η = ⇒ ( a − 1) d 2 ij ≥ a − η − 1 . Because a > 3 / 2, a − 1 > 1 / 2, we obtain d 2 ij ≥ 1 − η a − 1 . Thus the densit y of 2-cycles in this regular pair is at least 1 − η a − 1 . T o turn this pair into a complete bidirectional pair we ma y need to add at most 2 η a − 1 ℓ 2 arcs (b oth directions). The num b er of such go o d pairs is at most  m 2  ≈ m 2 2 , and each con tributes at most that many c hanges. Hence the total n umber of mo difications needed in this class is at most r − 1 r · η a − 1 n 2 (the factor r − 1 r accoun ts for the prop ortion of cross pairs). Summing these estimates, the total num b er of arcs that ha ve to b e changed is at most  2 ε + 2( d + ε ) + 2 η + r − 1 r · η a − 1  n 2 + o ( n 2 ) . 4.7 Choice of constants No w we c ho ose the constan ts in the following order: • Fix β > 0 as given. • Cho ose η > 0 so small that 2 η + r − 1 r · η a − 1 < β 10 . • Pic k d = β 20 and then ε ≤ min { ε 0 , β 20 } . • Finally c ho ose γ > 0 (in Theorem 4.1) sufficien tly small so that the δ in (2) is smaller than the δ required b y Lemma 4.2 for the c hosen η ; this is p oss ible b ecause δ tends to 0 as γ , ε, d → 0. With these choices the total num b er of changes is less than β n 2 for all sufficiently large n . T his completes the proof of Theorem 4.1. 5 T ypical structure of T t r +1 -free orien ted graphs and digraphs W e no w combine the con tainer theorem (Theorem 2.3) with the stabilit y result to pro ve the main theorem. 5.1 Rough structure: almost all graphs are almost r -partite W e first show that all but a tiny fraction of T t r +1 -free graphs are close to b eing r -partite. Lemma 5.1. F or every k ≥ 2 and any α > 0 , ther e exists ε > 0 such that for al l sufficiently lar ge n : 10 • A l l but at most f ( n, T t k +1 )2 − εn 2 T t k +1 -fr e e oriente d gr aphs on n vertic es c an b e made k -p artite by changing at most αn 2 ar cs. • A l l but at most f ∗ ( n, T t k +1 )2 − εn 2 T t k +1 -fr e e digr aphs on n vertic es c an b e made k - p artite by changing at most αn 2 ar cs. Pr o of. W e prov e (i); (ii) is analogous with a = 2. Let a = log 3. Cho ose constan ts 1 /n 0 ≪ ε ≪ γ ≪ β ≪ α, 1 /k , and set ε ′ = 2 ε . Apply Theorem 2.3 to H = T t k +1 with parameters N = n , ε ′ , obtaining a con tainer family C . Let C 1 = { G ∈ C : e a ( G ) ≥ ex a ( n, T t k +1 ) − ε ′ n 2 } . By the container theorem, |C | ≤ 2 εn 2 , so the num ber of T t k +1 -free oriented graphs not con tained in an y G ∈ C 1 is at most |C | 2 ex a ( n,T t k +1 ) − ε ′ n 2 ≤ f ( n, T t k +1 )2 − εn 2 . No w take any G ∈ C 1 . By prop ert y (b) of con tainers, G contains at most ε ′ n ( k +1) t copies of T t k +1 . Apply the remo v al lemma (Lemma 2.4) to delete at most γ n 2 arcs and obtain a T t k +1 -free digraph G ′ with e a ( G ′ ) ≥ ex a ( n, T t k +1 ) − ( ε ′ + γ ) n 2 . By the stability theorem (Theorem 4.1) with β , we hav e G ′ = DT k ( n ) ± β n 2 . Hence G itself differs from DT k ( n ) by at most ( β + γ ) n 2 ≤ αn 2 arcs. T his completes the proof. 5.2 Exact structure: almost all graphs are r -partite In this subsection we upgrade the appro ximate result of Lemma 5.1 to an exact one. The argumen t is an inductive counting pro cedure that closely follo ws the one in [6, Section 5], with the only change that the forbidden subgraph is now T t r +1 instead of T r +1 . F or completeness we give the full details, adapting the notation and lemmas accordingly . Recall that t r ( n ) denotes the num b er of edges in the r -partite T ur´ an graph T u r ( n ). F or an r -partition Q = ( V 1 , . . . , V r ) of [ n ] and a digraph G on [ n ], an arc is called cr ossing if its endp oints lie in differen t parts of Q , and non-cr ossing otherwise. A partition Q is called optimal for G if it minimises the num b er of non-crossing arcs. Giv en parameters η , µ > 0, we define several classes of T t r +1 -free orien ted graphs (the digraph case is analogous and will b e discussed at the end). • F Q ( n, T t r +1 , η ) : all lab elled T t r +1 -free oriented graphs on [ n ] for whic h Q is an optimal r -partition and the n umber of non-crossing arcs with resp ect to Q is at most η n 2 . • F Q ( n, T t r +1 , η , µ ) : all G ∈ F Q ( n, T t r +1 , η ) that additionally satisfy (F1) the num ber of non-crossing arcs is at most η n 2 ; (F2) for all distinct i, j ∈ [ r ] and all subsets U i ⊆ V i , U j ⊆ V j with | U i | , | U j | ≥ µn , − → e ( U i , U j ) , − → e ( U j , U i ) ≥ 1 6 | U i || U j | ; (F3) || V i | − n/r | ≤ µn for ev ery i ∈ [ r ]. • F ′ Q ( n, T t r +1 , η ) : the set of graphs in F Q ( n, T t r +1 , η ) that contain at least one non-crossing arc with resp ect to Q . Define the corresp onding cardinalities f Q , f Q ( η , µ ), f ′ Q in the natural w ay . The following lemma sho ws that the additional regularit y conditions (F2)–(F3) are satisfied by almost all graphs in F Q ( n, T t r +1 , η ). Lemma 5.2 (Go o d subfamily) . L et r ≥ 2 and let η , µ ∈ (0 , 1) satisfy µ 2 ≥ 24 H ( η ) , wher e H ( p ) = − p log p − (1 − p ) log(1 − p ) is the binary entr opy function. Then for al l sufficiently lar ge n and for every r -p artition Q of [ n ] , f Q ( n, T t r +1 , η ) − f Q ( n, T t r +1 , η , µ ) ≤ 3 t r ( n ) − µ 2 n 2 / 100 . 11 Pr o of. W e count the graphs that fail to satisfy (F3) or (F2). F or graphs failing (F3), by Prop osition 4.2 (the standard estimate on T ur´ an graphs) the n umber of p ossible cross- ing arcs is at most t r ( n ) − µ 2 n 2 / 3, while the n umber of c hoices for non-crossing arcs (at most η n 2 ) is at most 2 H ( η ) n 2 . Hence the contribution from (F3) failure is at most 2 H ( η ) n 2 3 t r ( n ) − µ 2 n 2 / 3 . F or graphs satisfying (F3) but failing (F2), consider a random orien ted graph where eac h crossing arc is c hosen uniformly from the three p ossibilities (no arc, forward, back- w ard). The probabilit y that a fixed pair ( U i , U j ) violates the density condition is, by Cher- noff ’s bound, at most 2 exp( −| U i || U j | / 8) ≤ 2 exp( − µ 2 n 2 / 8). There are at most 2 2 n c hoices for suc h subsets, so b y the union b ound and using µ 2 ≥ 24 H ( η ) w e obtain the desired b ound. A detailed calculation identical to that in [6, Lemma 5.2] yields the lemma. The next prop osition allows us to embed man y disjoint copies of T r in to an y graph satisfying (F2). It is a straigh tforward consequence of the regularity conditions and the greedy embedding lemma (Lemma 2.2); see [6, Proposition 5.3] for a pro of. Prop osition 5.3. L et n, r ∈ N , η , µ > 0 , let Q = ( V 1 , . . . , V r ) b e an r -p artition, and supp ose G ∈ F ∗ Q ( n, T t r +1 , η , µ ) (the digr aph version; the oriente d c ase is analo gous). F or every i ∈ [ r ] let B i ⊆ V i with | B i | ≥ 12 r − 2 µn . L et σ b e a p ermutation of [ r ] . Then G c ontains a c opy of T r on vertic es v 1 , . . . , v r such that v i ∈ B i and for al l distinct i, j , the ar c is dir e cte d fr om v i to v j iff σ ( i ) < σ ( j ) . Using this, we obtain a b ound on the num b er of non-crossing neigh b ours of an y vertex. Lemma 5.4. L et n, r ≥ 2 , η , µ > 0 , Q an r -p artition, and G ∈ F ∗ Q ( n, T t r +1 , η , µ ) . Then for every i ∈ [ r ] and every x ∈ V i , | N + V i ( x ) | + | N − V i ( x ) | ≤ 12 r − 2 · 2 µn. Pr o of. Assume the con trary . Then for some i and x , the internal degree exceeds 12 r − 2 2 µn . Because Q is optimal, for every other part j we m ust hav e at least as many neighbours in V j (otherwise moving x would reduce non-crossing arcs). Hence for each j we can select a set B j ⊆ N + V j ( x ) ∪ N − V j ( x ) of size 12 r − 2 µn , choosing the larger of the out- and in-neigh b ourho o ds. By Prop osition 5.3, there exists a cop y of T r with one v ertex in each B j and with a prescrib ed p ermutation. Adding x then yields a T r +1 , contradicting T t r +1 - freeness (since T t r +1 con tains T r +1 as a subgraph). See [6, Lemma 5.4] for full details. When w e remo ve t wo vertices x, y from the same part, the optimal partition of the remaining graph can v ary only a little. Lemma 5.5 (F ew optimal partitions) . L et r ≥ 2 , 0 < µ < 1 / (3 r 2 ) 12 , 0 < η < µ 2 / 3 , and let n b e sufficiently lar ge. Fix an r -p artition Q = ( V 1 , . . . , V r ) of [ n ] and two distinct vertic es x, y ∈ V 1 . Then ther e exists a set P of r -p artitions of [ n ] \ { x, y } , with |P | ≤ e µ 2 / 3 n , such that for every G ∈ F ∗ Q ( n, T t r +1 , η , µ ) , every optimal r -p artition of G − { x, y } b elongs to P . The pro of uses the fact that any tw o optimal partitions cannot differ to o muc h; oth- erwise (F2) w ould force man y non-crossing arcs. The details are exactly as in [6, Lemma 5.5], with k replaced b y r and T k +1 b y T t r +1 . W e are now ready to state the key coun ting lemma. 12 Lemma 5.6 (Main counting lemma) . F or every r ≥ 2 , ther e exist c onstants η > 0 and C > 0 (dep ending only on r ) such that for al l n and every r -p artition Q of [ n ] , f ′ Q ( n, T t r +1 , η ) ≤ C · 3 t r ( n ) · 2 − η n . The same holds for digr aphs with 3 r eplac e d by 4 . Pr o of. W e give the pro of for orien ted graphs; the digraph case is iden tical with 3 changed to 4 in all estimates. Cho ose constants 1 /C ≪ 1 /n 0 ≪ ε ≪ η ≪ µ ≪ 1 /r satisfying µ 2 ≥ 24 H ( η ) and η < µ 2 / 3. The pro of pro ceeds by induction on n . In fact w e sim ultaneously pro v e the stronger statement f Q ( n, T t r +1 ) ≤ 3 t r ( n ) (1 + C 2 − η n ) for all Q. (5.2) The base case n < n 0 is trivial b ecause 1 /C ≪ 1 /n 0 mak es the right-hand side larger than the total num b er of graphs. Assume n ≥ n 0 and that (5.2) holds for all smaller v alues. Fix a partition Q = ( V 1 , . . . , V r ). W e first b ound f ′ Q ( n, T t r +1 , η , µ ), i.e. the num ber of graphs in F Q ( n, T t r +1 , η , µ ) that contain at least one non-crossing arc. Let G ∈ F ′ Q ( n, T t r +1 , η , µ ) and pick a non-crossing arc xy ∈ V 1 (an y part will do; we fix V 1 for conv enience). W e count the n umber of p ossibilities in four steps. Step 1 – choosing xy . There are at most n 2 c hoices. Step 2 – the rest of the graph after deleting { x, y } . By Lemma 5.5, the optimal partitions of G − { x, y } b elong to a set P of size at most e µ 2 / 3 n . Applying the induction h yp othesis (5.2) to each such partition, we obtain at most X Q ′ ∈P f Q ′ ( n − 2 , T t r +1 ) ≤ e µ 2 / 3 n · 3 t r ( n − 2) (1 + C 2 − η ( n − 2) ) ≤ 3 t r ( n − 2) C e µ 1 / 2 n c hoices for the subgraph on [ n ] \ { x, y } . Step 3 – arcs b et ween { x, y } and v ertices outside V 1 . Let U b e the set of arcs c hosen in Step 2. Remov e from U all arcs inciden t to V 1 ; the remainder U ′ induces a subgraph G ′ on S r j =2 V j that b elongs to F ˜ Q ( n − | V 1 | , 5 η , 3 µ ) for a suitable partition ˜ Q (the restriction of Q to the other parts). By rep eatedly applying Prop osition 5.3 to G ′ , w e can find at least n/r − µn − 12 r − 3 3 µn vertex-disjoin t copies of T r − 1 , each with exactly one vertex in each V j ( j ≥ 2). F or eac h such copy K , consider the 2( r − 1) p otential arcs b etw een { x, y } and the vertices of K . T o k eep the whole graph T t r +1 -free, not all of the 3 2( r − 1) p ossible configurations are allo wed: a simple case analysis sho ws that at most 3 2( r − 1) − 1 of them are admissible, otherwise one could combine K with x, y to create a T t r +1 (the argument uses that T t r +1 is a blow-up of T r +1 and that x, y are in the same part). Moreov er, the n umber of v ertices outside V 1 not b elonging to any of these disjoin t copies is at most µ 1 / 2 n (b y the c hoice of constants). Hence the n umber of w ays to choose the arcs from { x, y } to the outside is b ounded b y  3 2( r − 1) − 1  n/r · 3 2 µ 1 / 2 n ≤ 3 2( r − 1) r n  1 − 3 − 2 r  n/r e µ 1 / 2 n ≤ 3 2( r − 1) r n e − n/ (9 r r ) e µ 1 / 2 n . Step 4 – arcs b etw een { x, y } and the remaining vertices of V 1 . By Lemma 5.4, eac h of x, y has at most 12 r − 2 2 µn neigh b ours inside V 1 . The n umber of wa ys to choose these neighbours and orien t the arcs is at most  n 12 r − 2 2 µn  2 · 2 2 · 12 r − 2 2 µn ≤ e µ 1 / 2 n , 13 using standard estimates and 1 /n 0 ≪ µ . Multiplying the four b ounds and using t r ( n ) ≥ t r ( n − 2) + 2( r − 1) r ( n − 2) (since adding t wo vertices to a balanced r -partite graph increases the n um b er of crossing arcs by roughly 2( r − 1) r n ), we obtain f ′ Q ( n, T t r +1 , η , µ ) ≤ n 2 · 3 t r ( n − 2) C e µ 1 / 2 n · 3 2( r − 1) r n e − n/ (9 r r ) · e µ 1 / 2 n ≤ 3 t r ( n ) C e − n/ (10 r r ) ≤ 3 t r ( n ) C 2 − 3 η n , where the last inequalit y uses that η is chosen small enough so that 2 − 3 η n ≥ e − n/ (10 r r ) . No w observe that f ′ Q ( n, T t r +1 , η ) = | F ′ Q ( n, T t r +1 , η , µ ) | + | F ′ Q ( n, T t r +1 , η ) \ F Q ( n, T t r +1 , η , µ ) | ≤ f ′ Q ( n, T t r +1 , η , µ )+  f Q ( n, T t r +1 , η ) − f Q ( n, T t r +1 , η , µ )  . Applying Lemma 5.2 to b ound the second term and the estimate for f ′ Q ( n, T t r +1 , η , µ ) just obtained, we get f ′ Q ( n, T t r +1 , η ) ≤ 3 t r ( n ) C 2 − 3 η n + 3 t r ( n ) − µ 2 n 2 / 100 ≤ 3 t r ( n ) C 2 − η n , b ecause µ 2 / 100 is m uch larger than η (recall η ≪ µ ). This prov es the desired b ound on f ′ Q . It remains to v erify (5.2). The n umber of graphs in F Q ( n, T t r +1 , η , µ ) that hav e no non-crossing arcs is at most 3 t r ( n ) (all crossing arcs can b e chosen arbitrarily). Hence f Q ( n, T t r +1 , η , µ ) − f ′ Q ( n, T t r +1 , η , µ ) ≤ 3 t r ( n ) . T ogether with the b ound on f ′ Q this yields f Q ( n, T t r +1 , η , µ ) ≤ 3 t r ( n ) (1 + C 2 − 3 η n ). Using Lemma 5.2 again, f Q ( n, T t r +1 , η ) ≤ 3 t r ( n ) (1 + C 2 − 3 η n ) + 3 t r ( n ) − µ 2 n 2 / 100 ≤ 3 t r ( n ) (1 + C 2 − 2 η n ) . Finally , from Lemma 5.1 (the rough structure) we hav e f ( n, T t r +1 ) − f ( n, T t r +1 , η ) ≤ f ( n, T t r +1 )2 − εn 2 ≤ 2 f ( n, T t r +1 , η )2 − εn 2 , which together with the b ound on f ( n, T t r +1 , η ) (obtained by summing ov er all partitions Q ) giv es f ( n, T t r +1 , η ) ≤ 3 t r ( n ) (1 + C 2 − 2 η n ) and consequen tly f Q ( n, T t r +1 ) ≤ 3 t r ( n ) (1 + C 2 − η n ), completing the induction. With Lemma 5.6 established, w e can no w finish the pro of of Theorem 1.2. Completion of The or em 1.2. Let η b e the constant from Lemma 5.6. By Lemma 5.1 there exists ε > 0 such that f ( n, T t r +1 ) ≤ f ( n, T t r +1 , η )(1 + 2 − εn 2 ) , where f ( n, T t r +1 , η ) = P Q f Q ( n, T t r +1 , η ) and the sum runs ov er all r -partitions Q of [ n ]. F or each Q , we ha ve f Q ( n, T t r +1 , η ) = f ′ Q ( n, T t r +1 , η ) + T Q ( n, r ), where T Q ( n, r ) is the n um- b er of r -partite orien ted graphs that hav e Q as an r -partition (i.e., all arcs are crossing). Clearly P Q T Q ( n, r ) = T ( n, r ). By Lemma 5.6 and the trivial b ound |{ Q }| ≤ r n , X Q f ′ Q ( n, T t r +1 , η ) ≤ r n · C 3 t r ( n ) 2 − η n = o  T ( n, r )  , since T ( n, r ) ≥ r n 3 t r ( n ) 2 r ! n r − 1 (a standard low er b ound, see e.g. [6, Lemma 5.1]). Therefore f ( n, T t r +1 ) ≤  T ( n, r ) + o ( T ( n, r ))  (1 + 2 − εn 2 ) = T ( n, r )(1 + o (1)) . The reverse inequalit y f ( n, T t r +1 ) ≥ T ( n, r ) is obvious b ecause every r -partite oriented graph is T t r +1 -free. Hence f ( n, T t r +1 ) = T ( n, r )(1 + o (1)), i.e. almost all T t r +1 -free oriented graphs are r -partite. The same argumen t with 4 instead of 3 giv es the digraph coun t f ∗ ( n, T t r +1 ) = T ∗ ( n, r )(1 + o (1)), completing the pro of of Theorem 1.2. 14 6 Concluding remarks and op en problems W e ha v e sho wn that for any r ≥ 2, t ≥ 1, and an y w eigh t parameter a ∈ (3 / 2 , 2], almost all T t r +1 -free oriented graphs and digraphs are r -partite. This generalises the KOTZ theorem and confirms a conjecture of Liang and Liu [7]. Our pro of relies on a w eighted extremal theorem, a stability result, and the container metho d. The restriction a > 3 / 2 is technical and comes from certain inequalities in the stabilit y proof; it w ould b e in teresting to extend the result to all a ≥ 1. Another natural direction is to consider blow-ups of other tournaments or cycles. The metho ds developed here should apply as long as the forbidden digraph has a suitable extremal structure (e.g., a complete r -partite digraph is extremal). F or cycles, the situation is more complex and already studied in [6]. References [1] N. Alon, A. Shapira, T esting subgraphs in directed graphs, J. Comput. System Sci. 69 (2004) 354–382. [2] W.G. Bro wn, F. Harary , Extremal digraphs, in: Combinatorial The ory and its Ap- plic ations , Coll. Math. So c. J. Boly ai 4 (1970) 135–198. [3] G. 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