Typical Structure of Oriented Graphs and Digraphs with Forbidden Blow-Up Transitive Triangles

Abstract

Transitive tournament (including transitive triangle) and its blow-up have some symmetric properties. In this work, we establish an analogue result of the Erdös-Stone theorem of weighted digraphs with a forbidden blow-up of the transitive tournament. We give a stability result of oriented graphs and digraphs with forbidden blow-up transitive triangles and show that almost all oriented graphs and digraphs with forbidden blow-up transitive triangles are almost bipartite, which reconfirms and strengthens the conjecture of Cherlin.

1. Introduction

Given a fixed graph H, a graph is called H-free if it does not contain a subgraph isomorphic to H. For brevity’s sake, other mentioned notations in the section are provided later in Section 2. In the study history of extremal graph theory, there are two types of important problems: (1) What are the maximum edges among H-free graphs on n vertices? (2) What is the typical structure of H-free graphs on n vertices? It is natural to consider graphs with some symmetry property, for example, the complete graphs $K_r$, cycles $C_k$ on r vertices, respectively. The significant progress of the first problem was made by Turán in 1941 who determined the maximum edges among $K_{r+1}$-free graphs on n vertices and the corresponding extremal graphs. In 1946 Erdös and Stone [1] extended Turán’s theorem by replacing the forbidden subgraph $K_{r+1}$ by its blow-up and asymptotically determined the maximum edges. The second problem started in 1976 when Erdös, Kleitman and Rothschild [2] showed that almost all $K_3$-free graphs are bipartite and asymptotically determined the logarithm of the number of $K_{r+1}$-free graphs on n vertices, for every integer $r\ge 2$. This was strengthened by Kolaitis, Prömel and Rothschild [3], who showed that almost all $K_{r+1}$-free graphs are r-partite, for every integer $r\ge 2$. These work inspired a vast body of works concerning the maximum edges, the number and structure of H-free graphs among H-free graphs respectively (see, e.g., [3,4,5,6,7,8,9,10,11]). More recently, some related results have been proved for hypergraphs (see, e.g., [12,13]).

All the works mentioned above dealt with undirected graphs. It is natural to generalize those results to digraphs or oriented graphs. The study of the first problem of digraphs and oriented graphs started by Brown and Harary [14] in 1970. They considered and determined the n-vertex digraphs with maximum edges and not containing the transitive tournament $T_{r+1}$ in [14]. In 2017, Kühn, Osthus, Townsend and Zhao [15] extended this result to weighted digraphs. However, the similar result of the Erdös and Stone [1] theorem for weighted digraphs is still open. In this work, we will establish an analogue Erdös and Stone result of weighted digraphs.

For the study of the second problem of digraphs and oriented graphs, Cherlin [16] gave a classification of countable homogeneous oriented graphs. He remarked that ‘the striking work of [3] does not appear to go over to the directed case’ and conjectured that almost all $T_3$-free oriented graphs are tripartite in 1998. Kühn, Osthus, Townsend and Zhao [15] verified this conjecture and showed that almost all $T_{r+1}$-free oriented graphs and almost all $T_{r+1}$-free digraphs are r-partite.

It is natural to ask a similar question: what are the typical structures of digraphs and oriented graphs not containing the blow-up of $T_{r+1}$? In this work, we shall reconfirm and generalize Cherlin’s conjecture [15]. We show that almost all $T_3^t$-free oriented graphs and almost all $T_3^t$-free digraphs are almost bipartite for any positive integer t, where $T_3^t$ is the blow-up of the transitive triangle $T_3$.

The rest of the paper is organized as followed. Some notations and useful tools and our results will be laid out in Section 2. The Regularity Lemma of digraphs and the proof of our first result will be given in Section 3. A stability result of digraphs and the proof of our second result will be given in Section 4. The concluding remarks will be given in Section 5.

2. Notations, Tools and Results

We shall give some notions before we start to state some relevant results. A digraph is a pair $(V,E)$ where V is a set of vertices and E is a set of ordered pairs of distinct vertices in V (note that this means we do not allow loops or multiple arcs in the same direction in a digraph). An oriented graph is a digraph with at most one arc between two vertices, so may be considered as an orientation of a simple undirected graph.

Given a class of graphs (digraphs or oriented graphs, respectively) $\mathcal{A}$, we let ${\mathcal{A}}_n$ denote the set of all graphs (digraphs or oriented graphs, respectively) in $\mathcal{A}$ that have precisely n vertices, and we say that almost all graphs (digraphs or oriented graphs, respectively) in $\mathcal{A}$ have property $\mathcal{B}$ or, the typical structure of $\mathcal{A}$ is $\mathcal{B}$ if

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\frac{\left|\right\{G\in {\mathcal{A}}_n:G\mathrm{has}\mathrm{property}\mathcal{B}\left\}\right|}{|{\mathcal{A}}_n|}=1.\end{array}$$

Let $G=(V,E)$ be a digraph, we write $uv$ for the arc directed from vertex u to vertex v. Denote by $N_G^{+}\left(v\right):=\{u\in V:vu\in E\}$ and $N_G^{-}\left(v\right):=\{u\in V:uv\in E\}$ the out-neighborhood and the in-neighborhood of v, respectively. Denote by $d_G^{+}\left(v\right):=\left|N_G^{+}\left(v\right)\right|$ and $d_G^{-}\left(v\right):=\left|N_G^{-}\left(v\right)\right|$ the out-degree and the in-degree $d_G^{-}\left(v\right)$ of v, respectively. Denote by $N_G\left(v\right):=N_G^{-}\bigcup N_G^{+}$ and $N_G^{\pm }\left(v\right):=N_G^{-}\bigcap N_G^{+}$ the neighborhood and the intersection neighborhood of v. Denote by $\Delta \left(G\right),{\Delta }^{+}\left(G\right)$ and ${\Delta }^{-}\left(G\right)$ the maximum of $|N_G\left(v\right)|,|N_G^{+}\left(v\right)|$ and $|N_G^{-}\left(v\right)|$ among all $v\in G$, respectively. Denote by ${\Delta }^0\left(G\right)$ the maximum of $d^{+}\left(v\right)$ and $d^{-}\left(v\right)$ among all $v\in V$. For $A\subset V\left(G\right)$, denote by $G\left[A\right]$ and $G-A$ the sub-digraph of G induced by A and the digraph obtained from G by deleting all vertices in A and all arcs incident to A, respectively. Given two disjoint subsets A and B of vertices of G, an $A\to B$ arc is an arc $ab$ where $a\in A$ and $b\in B$. We denote by $E(A,B)$ for the set of all these arcs and $e_G(A,B):=\left|E(A,B)\right|$. Denote by ${(A,B)}_G$ the bipartite oriented subgraph of G whose vertex class are A and B and whose arc set is $E(A,B)$. The density of ${(A,B)}_G$ is defined to be

$$d_G(A,B):=\frac{e_G(A,B)}{\left|A\right|\left|B\right|}.$$

Given $ϵ>0$, we call ${(A,B)}_G$ an $ϵ$-regular pair if $\left|d\right(X,Y)-d(A,B\left)\right|<ϵ$ holds for all subsets $X\subseteq A$ and $Y\subseteq B$ with $\left|X\right|>ϵ\left|A\right|$ and $\left|Y\right|>ϵ\left|B\right|$. Note that $(B,A)$ may not necessarily be an $ϵ$-regular pair since the order matters.

Given a digraph $G=(V,E)$, denote by $f_1\left(G\right)$ the number of pairs $u,v\in V$ such that exactly one of $uv$ and $vu$ is an arc of G, and $f_2\left(G\right)$ the number of pairs $u,v\in V$ such that both $uv$ and $vu$ are arcs of G(we call $uv$ a double edge for convenience). For a vertex v, denote by $f_1\left(v\right)$ the number $u\in V$ such that exactly one of $uv$ and $vu$ is an arc of G, and $f_2\left(v\right)$ the number double edges incident with v. For a real number $a\ge 1$, the weighted size of G is defined by $e_a\left(G\right):=a·f_2\left(G\right)+f_1\left(G\right)$. For convenience, we write $e\left(H\right):=e_a\left(H\right)$ for oriented graph H. For a vertex v, its weight is defined by $e_a\left(v\right):=a·f_2\left(v\right)+f_1\left(v\right).$ This definition allows for a unified approach to extremal problems on undirected graphs, oriented graphs and digraphs. Because for an undirected graph, $e\left(G\right):=e_1\left(G\right)$ is the number of its edges when we equate an undirected edge in an undirected graph with a double edge in a digraph. Furthermore, a digraph G contains $4^{f_2\left(G\right)}{2}^{f_1\left(G\right)}=2^{e_2\left(G\right)}$ labelled sub-digraphs and $3^{f_2\left(G\right)}{2}^{f_1\left(G\right)}=2^{e_{log3}\left(G\right)}$ labelled oriented subgraphs if we set $a=2$ and $a=log3$, respectively.

The Turán graph $T{u}_r\left(n\right)$ is an undirected graph of n vertices which is formed by partitioning the set of n vertices into r-parts of nearly equal size, and connecting two vertices by an edge whenever they belong to different parts. Denote by $t_r\left(n\right)$ the edge size (number of edges) of $T{u}_r\left(n\right)$. Denote by $D{T}_r\left(n\right)$ the digraph obtained from $T{u}_r\left(n\right)$ by replacing each undirected edge of $T{u}_r\left(n\right)$ with a double edge. Denote by $D{K}_r$ the digraph obtained from the complete graph $K_r$ on r vertices by replacing each edge of $K_r$ by a double edge.

Given a digraph H, the weighted Turán number $e{x}_a(n,H)$ is the maximum $e_a\left(G\right)$ among all H-free digraphs G on n vertices. It is easy to see that $D{T}_r\left(n\right)$ is $T_{r+1}$-free, so $e{x}_a(n,T_{r+1})\ge e_a\left(D{T}_r\left(n\right)\right)=a·t_r\left(n\right)$.

A transitive oriented graph is an oriented graph such that whenever it contains arcs $uv$ and $vw$ then it contains $uw$ too. A transitive tournament $T_n$ on n vertices is the orientation of $K_n$ such that it is transitive. Denote by $T_{r+1}^t$ the blow-up of $T_{r+1}$ for some positive integer t by replacing every vertex $v_i$ of $T_{r+1}$ with an independent set of t vertices and connecting every pair of vertices whenever they belong to different independent sets with the homogeneous direction in accordance with that of $T_{r+1}$. For a positive integer k we write $\left[k\right]:=\{1,\dots ,k\}$. For convenience, we drop the subscripts of all notions if they are unambiguous.

We need the following result of the forbidden digraphs container of Kühn et al. [15], which allows us to reduce an asymptotic counting problem to an extremal problem. Given an oriented graph H with $e\left(H\right)\ge 2$, we let

$$m\left(H\right)=\underset{H^{\prime }\subset H,e\left(H^{\prime }\right)>1}{max}\frac{e\left(H^{\prime }\right)-1}{v\left(H^{\prime }\right)-2}.$$

Theorem 1

(Theorem 3.3 [15]). Let H be an oriented graph with $h:=v\left(H\right)$ and $e\left(H\right)\ge 2$, and let $a\in \mathbb{R}$ with $a\ge 1$. For every $ϵ>0$, there exists $c>0$ such that for all sufficiently large N, there exists a collection $\mathcal{C}$ of digraphs on the vertex set $\left[n\right]$ with the following properties.

(a) 

For every H-free digraph I on $\left[N\right]$ there exists $G\in \mathcal{C}$ such that $I\subset G$.

(b) 

Every digraph $G\in \mathcal{C}$ contains at most $ϵN^h$ copies of H, and $e_a\left(G\right)\le e{x}_a(N,H)+ϵN^2$.

(c) 

$log\left|\mathcal{C}\right|\le c{N}^{2-1/m\left(H\right)}logN$.

Note that this result is essentially a consequence of a recent and very powerful result of Balogh, Morris and Samotij [7] and Saxton and Thomason [17], which introduces the notion of hypergraph containers to give an upper bound on the number of independent sets in hypergraphs, and a digraph analogue [15] of the well-known supersaturation result of Erdös and Simonovits [1].

As mentioned in Section 1, Brown and Harary [14] first determined the extremal digraph with the maximum edge among $T_{r+1}$-free digraphs with n vertices. Kühn, Osthus, Townsend and Zhao [15] extended this result to weighted digraphs.

Lemma 1

([15]). Let $a\in (\frac{3}{2},2]$ be a real number and let $r,n\in \mathbb{N}$. Then $e{x}_a(n,T_{r+1})=a·t_r\left(n\right)$, and $D{T}_r\left(n\right)$ is the unique extremal $T_{r+1}$-free digraph on n vertices.

We generalize the above result in the following.

Theorem 2.

For all positive integers $r,t$, every real numbers $a\in (\frac{3}{2},2]$ and $\gamma >0$, there exists an integer $n_0$ such that every digraph G with $n\ge n_0$ vertices and

$$e_a\left(G\right)\ge a·t_r\left(n\right)+\gamma n^2$$

contains $T_{r+1}^t$ as a sub-digraph.

Example 1.

For a given digraph H, it is contained in its blow-up $H^t$ for each positive integer t. Thus a H-free digraph (oriented graph) G is also a $H^t$-free digraph (oriented graph), but a $H^t$-free digraph (oriented graph) is not necessarily a H-free digraph (oriented graph). For example, $T_{r+1}$ is a sub-digraph of $T_{r+1}^t$ and a $T_{r+1}$-free digraph (oriented graph) G is also a $T_{r+1}^t$-free digraph (oriented graph), but a $T_{r+1}^t$-free digraph (oriented graph) is not necessary a $T_{r+1}$-free digraph (oriented graph). However, the above results show that the extremal weighted sizes are asymptotically equal. One can also see that there are many other sub-digraphs contained in $T_{r+1}^t$, for example, the stars with the orientation all going away from the centre or going toward the centre, respectively. For those digraph-free digraphs, the corresponding extremal problems are still open. Our result may shed some light on them.

In 1998 Cherlin [16] gave the following conjecture.

Conjecture 1

(Cherlin [16]). Almost all $T_3$-free oriented graphs are tripartite.

Kühn, Osthus, Townsend and Zhao [15] verified this conjecture and showed that almost all $T_{r+1}$-free oriented graphs and almost all $T_{r+1}$-free digraphs are r-partite. We will strengthen Cherlin’s conjecture and show that almost all $T_3^t$-free oriented graphs and almost all $T_3^t$-free digraphs are almost bipartite for any positive integer t. Let $f(n,T_{r+1}^t)$ and $f^{*}(n,T_{r+1}^t)$ denote the number of labelled $T_{r+1}^t$-free oriented graphs and digraphs on n vertices, respectively.

Theorem 3.

For every positive integer $t\in \mathbb{N}$ and any $\alpha >0$ there exists $ϵ>0$ such that the following holds for all sufficiently large n.

(i) 

All but at most $f(n,T_3^t)2^{-ϵn^2}$$T_3^t$-free oriented graphs on n vertices can be made bipartite by changing at most $\alpha n^2$ edges.

(ii) 

All but at most $f^{*}(n,T_3^t)2^{-ϵn^2}$$T_3^t$-free digraphs on n vertices can be made bipartite by changing at most $\alpha n^2$ edges.

3. The Regularity Lemma and the Proof of Theorem 2

In this section, we shall give the proof of Theorem 2. First, we need the regularity lemma of digraphs of Alon and Shapira [18]. See [19] for a survey on the Regularity Lemma.

Given partitions $V_0,V_1,\dots ,V_k$ and $U_0,U_1,\dots ,U_{\ell }$ of the vertex set of a digraph, we say that $V_0,V_1,\dots ,V_k$ refines $U_0,U_1,\dots ,U_{\ell }$ if for all $V_i$ with $i\ge 1$ there is some $U_j$ with $j\ge 0$ such that $V_i\subseteq U_j$. Note that $V_0$ need not be contained in any $U_j$.

Lemma 2

(Degree form of the Regularity Lemma of Digraphs [18]). For every $ϵ\in (0,1)$ and all integers $M^{\prime },M^{″}$ there are integers M and $n_0$ such that if

• G is a digraph on $n\ge n_0$ vertices,
• $U_0,\dots ,U_{M^{″}}$ is a partition of the vertex set of G,
• $d\in [0,1]$ is any real number,

then there is a partition of the vertex set of G into $V_0,\dots ,V_k$ and a spanning sub-digraph $G^{\prime }$ of G such that the following holds:

(1) 

$M^{\prime }\le k\le M$,

(2) 

$|V_0|\le ϵ·n$,

(3) 

$|V_1|=\dots =|V_k|=\ell$,

(4) 

$V_0,\dots ,V_k$ refines the partition $U_0,\dots ,U_{M^{″}}$,

(5) 

$d_{G^{\prime }}^{+}\left(x\right)>d_G^{+}\left(x\right)-(d+ϵ)n$ for all vertices x of G,

(6) 

$d_{G^{\prime }}^{-}\left(x\right)>d_G^{-}\left(x\right)-(d+ϵ)n$ for all vertices x of G,

(7) 

$G^{\prime }\left[V_i\right]$ is empty for all $i=1,\dots ,k$,

(8) 

the bipartite oriented graph ${(V_i,V_j)}_{G^{\prime }}$ is ϵ-regular and has density either 0 or density at least d for all $1\le i,j\le k$ and $i\ne j$.

We call $V_1,\dots ,V_k$ clusters and $V_0$ the exceptional set. The last condition of the lemma says that all pairs of clusters are $ϵ$-regular in both directions (but possibly with different densities). We call the spanning digraph $G^{\prime }\subseteq G$ in the lemma the pure digraph with parameters $ϵ,d,\ell$. Given clusters $V_1,\dots ,V_k$ and a digraph $G^{\prime }$, the reduced digraph R with parameters $ϵ,d,\ell$ is the digraph whose vertices are $V_1,\dots ,V_k$ and whose arcs are all the $V_i\to V_j$ arcs in $G^{\prime }$ that is $ϵ$-regular and has a density of at least d.

Note that a simple consequence of the $ϵ$-regular pair $(A,B)$: for any subset $Y\subseteq B$ that is not too small, most vertices of A have about the expected number of out-neighbours in Y; and similarly, for any subset $X\subseteq A$ that is not too small, most vertices of B have about the expected number of in-neighbours in X.

Lemma 3.

Let $(A,B)$ be an ϵ-regular pair, of density d say, and $X\subseteq A$ has size $\left|X\right|\ge ϵ\left|A\right|$ and $Y\subseteq B$ has size $\left|Y\right|\ge ϵ\left|B\right|$. Then all but at most $ϵ\left|A\right|$ of vertices in A have (each) at least $(d-ϵ)\left|Y\right|$ out-neighbors in Y and all but at most $ϵ\left|B\right|$ of vertices in B have (each) at least $(d-ϵ)\left|X\right|$ in-neighbors in X.

Proof. 

Let $A^{\prime }$ be the set of vertices with out-neighbors in Y less than $(d-ϵ)\left|Y\right|$. Then $e(A^{\prime },Y)<|A^{\prime }|(d-ϵ)\left|Y\right|$, so

$$d(A^{\prime },Y)=\frac{e(A^{\prime },Y)}{|A^{\prime }\left|\right|Y|}<d-ϵ=d(A,B)-ϵ.$$

Since $(A,B)$ is $ϵ$-regular, this implies that $|A^{\prime }|<ϵ|A|$.

Similarly, let $B^{\prime }$ be the set of vertices with in-neighbours in X less than $(d-ϵ)\left|X\right|$. Then $e(X,B^{\prime })<\left|X\right|(d-ϵ)\left|B^{\prime }\right|$, so

$$d(X,B^{\prime })=\frac{e(X,B^{\prime })}{\left|X\right||B^{\prime }|}<d-ϵ=d(X,B)-ϵ.$$

Since $(A,B)$ is $ϵ$-regular, this implies that $|B^{\prime }|<ϵ|B|$. □

The following lemma says that the blow-up $R^s$ of the reduced digraph R can be found in G, provided that $ϵ$ is small enough and the $V_i$ is large enough.

Lemma 4.

For all $d\in (0,1)$ and $\Delta \ge 1$, there exists an ${ϵ}_0>0$ such that if G is any digraph, s is an integer and R is a reduced digraph of $G^{\prime }$, where $G^{\prime }$ is the pure digraph of G with parameters $ϵ\le {ϵ}_0$, $\ell \ge s/{ϵ}_0$ and d. For any digraph H with $\Delta \left(G^{\prime }\right)\le \Delta$, then

$$H\subseteq R^s\Rightarrow H\subseteq G^{\prime }\subseteq G.$$

Proof. 

The proof is similar with that of Lemma 7.3.2 in [20]. Given d and $\Delta$, choose ${ϵ}_0<d$ small enough such that

$$\begin{array}{c}\hfill \frac{\Delta +1}{{(d-{ϵ}_0)}^{\Delta }}{ϵ}_0\le 1;\end{array}$$

(1)

such a choice is possible, since $\frac{\Delta +1}{{(d-ϵ)}^{\Delta }}ϵ\to 0$ as $ϵ\to 0$. Now let $G,H,s,R$ be given as stated. Let $\{V_0,V_1,\dots ,V_k\}$ be the $ϵ$-regular partition of $G^{\prime }$ that give rise to R. Thus, $ϵ<{ϵ}_0,V\left(R\right)=\{V_1,\dots ,V_k\}$ and $|V_1|=\dots =|V_k|=\ell$. Let us assume that H is actually a sub-digraph of $R^s$, with vertices $u_1,\dots ,u_h$. Each vertex $u_i$ lies in one of the s-sets $V_j^s$ of $R^s$. This defines a map $\sigma :i\mapsto j$. We aim to define an embedding $u_i\mapsto v_i\in V_{\sigma \left(i\right)}$ of H in $G^{\prime }$; thus, $v_1,\dots ,v_h$ will be distinct, and $v_i{v}_j$ will be an arc of $G^{\prime }$ whenever $u_i{u}_j$ is an arc of H.

We will choose the vertices $v_1,\dots ,v_h$ inductively. Throughout the induction, we shall have a “target set” $Y_i\subseteq V_{\sigma \left(i\right)}$ assigned to each i; this contains the vertices that are still candidates for the choice of $v_i$. Initially, $Y_i$ is the entire set $V_{\sigma \left(i\right)}$. As the embedding proceeds, $Y_i$ will get smaller and smaller (until it collapses to $\left\{v_i\right\}$): whenever we choose a vertex $v_j$ with $j<i$ and if

Case (i):

$u_i$ are both out-neighbor and in-neighbor of $u_j$ in H, we delete all those vertices from $Y_i$ that are not adjacent to $v_j$ with double edges.

Case (ii):

$u_i$ is just the out-neighbor of $u_j$ in H, we delete all those vertices from $Y_i$ that are not the out-neighbor of $v_j$.

Case (iii):

$u_i$ is just the in-neighbour of $u_j$ in H, we delete all those vertices from $Y_i$ that are not the in-neighbour of $v_j$.

To make this approach work, we have to ensure that the target set $Y_i$ does not get too small. When we come to embed a vertex $u_j$, we consider all the indices $i>j$ such that $u_i$ is adjacent to $u_j$ in H; there are at most $\Delta$ such i. For each of these i, we wish to select $v_j$ so that

$$\begin{array}{c}\hfill Y_i^j=N^{*}\left(v_j\right)\bigcap Y_i^{j-1}\end{array}$$

(2)

is large, where

$$\begin{array}{c}\hfill N^{*}\left(v_j\right)=\left\{\begin{array}{cc}{N}^{\pm }\left(v_j\right)\hfill & \mathrm{if}{u}_i\mathrm{are}\mathrm{both}\mathrm{out-neighbor}\mathrm{and}\mathrm{in-neighbor}\mathrm{of}{u}_j;\hfill \\ N^{+}\left(v_j\right)\hfill & \mathrm{if}{u}_i\mathrm{is}\mathrm{out-neighbor}\mathrm{of}{u}_j;\hfill \\ N^{-}\left(v_j\right)\hfill & \mathrm{if}{u}_i\mathrm{is}\mathrm{in-neighbor}\mathrm{of}{u}_j.\hfill \end{array}\right.\end{array}$$

Now this can be done by Lemma 3: unless $Y_i^{j-1}$ is tiny (of size less than $ϵ\ell$), all but at most $ϵ\ell$ choices of $v_j$ will be such that (2) implies

$$\begin{array}{c}\hfill |Y_i^j|\ge (d-ϵ)|Y_i^{j-1}|\end{array}$$

(3)

Doing this simultaneously for all of at most $\Delta$ values of i considered, we find that all but at most $\Delta ϵ\ell$ choices of $v_j$ from $V_{\sigma \left(j\right)}$, and in particular from $Y_j^{j-1}\subseteq V_{\sigma \left(j\right)}$, satisfy (3) for all i.

It remains to show that $|Y^{j-1}|-\Delta ϵ\ell \ge s$ to ensure that a suitable choice for $v_j$ exists: since $\sigma \left(j^{\prime }\right)=\sigma \left(j\right)$ for at most $s-1$ of the vertices $u_{j^{\prime }}$ with $j^{\prime }<j$, a choice between s suitable candidates for $v_j$ will suffice to keep $v_j$ distinct from $v_1,\dots ,v_{j-1}$. But all this follows from our choice of ${ϵ}_0$. Indeed, the initial target sets $Y_i^0$ have size ℓ, and each $Y_i$ has vertices deleted from it only when some $v_j$ with $j<i$ and $u_j$ and $u_i$ are adjacent in H, which happens at most $\Delta$ times. Thus,

$$|Y_i^j{|-\Delta ϵ\ell \ge (d-ϵ)}^{\Delta }-\Delta ϵ\ell \ge {(d-{ϵ}_0)}^{\Delta }-\Delta {ϵ}_0\ell \ge {ϵ}_0\ell \ge s$$

whenever $j<i$, so in particular $|Y_i^j|-\Delta \ge {ϵ}_0\ell \ge ϵ\ell$ and $|Y_j^{j-1}|-\Delta \ge ϵ\ell \ge s$. □

We can now prove Theorem 2 using Lemma 1, Lemma 4 and the Regularity Lemma of digraphs.

Proof of Theorem 2.

Let $d:=\gamma ,\Delta =\Delta \left(K_{r+1}^s\right)$, then Lemma 4 returns an ${ϵ}_0>0$. Assume

$$\begin{array}{c}\hfill {ϵ}_0<\gamma /2<1\end{array}$$

(4)

Let $M^{\prime },M^{″}>1/\gamma$, choose $ϵ>0$ small enough that $ϵ\le {ϵ}_0$ and $\delta :=(a-1)d-ϵ-a{ϵ}^2/2-aϵ>0$. The Regularity Lemma of digraphs returns an integer M. Assume

$$n\ge \frac{Ms}{{ϵ}_0(1-ϵ)},$$

Since $\frac{Ms}{{ϵ}_0(1-ϵ)}\ge M^{\prime },M^{″}$. Let $\{U_0,U_1,\cdots ,U_{M^{″}}\}$ be a partition of vertex set of G. The Regularity Lemma of digraphs provided us with an $ϵ-$regular partition $\{V_0,V_1,\dots ,V_k\}$ of $G^{\prime }$, the pure digraph of G, with parameters $ϵ,d,\ell$ and $M^{\prime }\le k\le M$. That is $|V_1|=\dots =|V_k|=\ell$ and $|V_0|<ϵn$. Then

$$\begin{array}{c}\hfill n\ge k\ell \end{array}$$

(5)

$$\ell =\frac{n-|V_0|}{k}\ge \frac{n-ϵn}{M}=n\frac{1-ϵ}{M}\ge \frac{s}{{ϵ}_0}$$

by the choice of n. Let R be the regularity digraph of $G^{\prime }$ with parameters $ϵ,\ell ,d$ corresponding to the above partition. Since $ϵ\le {ϵ}_0,\ell \ge s/{ϵ}_0$. R satisfies the premise of Lemma 4 and $\Delta \left(K_{r+1}^s\right)=\Delta$. Thus, to conclude by Lemma 4 that $T_{r+1}^s\subseteq G^{\prime }$, all that remains to be checked is that $T_{r+1}\subseteq R$.

Our plan was to show $T_{r+1}\subseteq R$ by Lemma 1. We thus have to check that the weight of R is large enough.

By (5) and (6) of the regularity Lemma of digraphs, we have

$$\begin{array}{c}\hfill {\Vert G\Vert }_a\le {\Vert G^{\prime }\Vert }_a+(d+ϵ)n^2\end{array}$$

(6)

At most $\left(\genfrac{}{}{0pt}{}{|V_0|}{2}\right)$ double edges lie inside $V_0$, and at most $|V_0|k\ell \le ϵnk\ell$ double edges join $|V_0|$ to other partition sets. The $ϵ-$regular pairs in $G^{\prime }$ of 0 density contribute nothing to the weight of $G^{\prime }$. Since each edge of R corresponds to at most ${\ell }^2$ edges of $G^{\prime }$, we thus have in total

$$\begin{array}{c}\hfill \Vert G^{\prime }{\Vert }_a\le \frac{1}{2}a{ϵ}^2{n}^2+aϵnk\ell +{\Vert R\Vert }_a{\ell }^2.\end{array}$$

Combined with (6), we get

$$\begin{array}{cc}\hfill {\Vert R\Vert }_a& \ge k^2·\frac{a(\frac{r-1}{r}+\gamma )n^2-(d+ϵ)n^2-\frac{1}{2}a{ϵ}^2{n}^2-aϵnk\ell }{k^2{\ell }^2}\hfill \\ \hfill & \ge a\frac{r-1}{r}{k}^2+\delta k^2\hfill \\ \hfill & =a·t_r\left(k\right)+\delta k^2\hfill \\ \hfill & >a·t_r\left(k\right),\hfill \end{array}$$

for all sufficiently large n. Therefore $T_{r+1}\subseteq R$ by Lemma 1, as desired. □

Similar to the Erdös-Stone theorem of undirected graphs, the Erdös-Stone theorem of digraphs is interesting not only in its own right, but also has an interesting corollary. For an oriented graph H, its chromatic number is defined as the chromatic number of its underlying graph. An oriented graph H with chromatic number $\chi \left(H\right)$ is called homogeneous if there is a colouring of its vertices by [$\chi \left(H\right)$] such that either $E(V_i,V_j)=\varnothing$ or $E(V_j,V_i)=\varnothing$ for every $1\le i\ne j\le \chi \left(H\right)$, where $V_i$ is the vertex set with colour i.

Given an acyclic homogeneously oriented graph H and an integer n, consider the number $h_n:=e{x}_a(n,H)/\left(a\left(\genfrac{}{}{0pt}{}{n}{2}\right)\right)$: the maximum weighted density that an $n-$vertex digraph can have without containing a copy of H.

Theorem 2 implies that the limit of $h_n$ as $n\to \infty$ is determined by a very simple function of a natural invariant of H–its chromatic number!

Corollary 1.

For every acyclic homogeneously oriented graph H with at least one edge,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\frac{e{x}_a(n,H)}{a\left(\genfrac{}{}{0pt}{}{n}{2}\right)}=\frac{\chi \left(H\right)-2}{\chi \left(H\right)-1}.\end{array}$$

Proof of Corollary 1.

Let $r:=\chi \left(H\right)$. Since H can not be coloured with $r-1$ colours, we have $H⊈D{T}_{r-1}\left(n\right)$ for all $n\in \mathbb{N}$, and hence

$$\begin{array}{c}\hfill a·t_{r-1}\left(n\right)\le e{x}_a(n,H).\end{array}$$

On the other hand, we have $H\subseteq T_r^t$ for all sufficiently large t. Thus

$$\begin{array}{c}\hfill e{x}_a(n,H)\le e{x}_a(n,T_r^t)\end{array}$$

for sufficiently large t. Fix such t, Theorem 2 implies that eventually (i.e., for large enough n)

$$\begin{array}{c}\hfill e{x}_a(n,T_r^t)<a{t}_{r-1}\left(n\right)+ϵn^2.\end{array}$$

Hence for large enough n,

$$\begin{array}{cc}\hfill \frac{t_{r-1}\left(n\right)}{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}& \le \frac{e{x}_a(n,H)}{a\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\hfill \\ \hfill & \le \frac{e{x}_a(n,T_r^t)}{a\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\hfill \\ \hfill & <\frac{t_{r-1}\left(n\right)}{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}+\frac{ϵn^2}{a\left(\genfrac{}{}{0pt}{}{n}{2}\right)}\hfill \\ \hfill & <\frac{t_{r-1}\left(n\right)}{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}+\frac{2ϵ}{a(1-1/n)}\hfill \\ \hfill & \le \frac{t_{r-1}\left(n\right)}{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}+4ϵ\hfill \end{array}$$

Since $\frac{t_{r-1}\left(n\right)}{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}$ converges to $\frac{r-2}{r-1}$ as $n\to \infty$, we get $\frac{e{x}_a(n,H)}{a\left(\genfrac{}{}{0pt}{}{n}{2}\right)}=\frac{r-2}{r-1},$ for every $ϵ>0$. □

4. Stability of Digraphs and Proof of Theorem 3

In this section, we will establish the stability of $T_{r+1}^t$-free digraphs and give a proof of Theorem 3. First, we need the following lemma.

Lemma 5.

Let G be a digraph obtained by adding a new vertex u and connecting it to each vertex of $D{K}_{r-1}$ with an arc, then G contains $T_r$.

Proof. 

We divide the vertices of $D{K}_{r-1}$ into two classes, i.e., $N_G^{-}\left(u\right)$ and $N_G^{+}\left(u\right)$. Then we can arbitrarily order the vertices of $N_G^{-}\left(u\right)$ and $N_G^{+}\left(u\right)$ respectively. Assume $|N_G^{-}\left(u\right)|=p$, $|N_G^{+}\left(u\right)|=r-p-1$, and $N_G^{+}\left(u\right)=\{v_1,\dots ,v_p\},N_G^{-}\left(u\right)=\{u_1,\dots ,u_{r-p-1}\}$. Then we order the vertices of G as $\{v_1,\dots ,v_p,u,u_1,\dots ,u_{r-p-1}\}$. A $T_r$ can be chosen by choosing the vertices and all their out-edges which connect to all the vertices behind it in this order. Obviously, the result is true if $p=0$ or $r-p-1=0$. □

We now give the result of stability of $T_{r+1}^t$-free digraphs.

Theorem 4.

(Stability) Let $t\in \mathbb{N}$ and $a\in \mathbb{R}$ with $t\ge 1,a\in (\frac{3}{2},2]$. Then for any $T_3^t$-free digraph G with

$$\begin{array}{c}\hfill e_a\left(G\right)=a(\frac{1}{2}+o\left(1\right))\frac{n^2}{2}\end{array}$$

satisfies $G=D{T}_2\left(n\right)\pm o\left(n^2\right)$.

Proof. 

We claim that there are $\mathsf{\Omega }\left(n^2\right)$ double edges in G. For otherwise, we assume that there are $o\left(n^2\right)$ double edges in G. Delete all double edges from G and denote the resulting digraph by $G^{\prime }$. Then the digraph $G^{\prime }$ is $T_3^t$-free and contains no double edges. And $e_a\left(G^{\prime }\right)=a(\frac{1}{2}+o\left(1\right))\frac{n^2}{2}-a·o\left(n^2\right)=a(\frac{1}{2}+o\left(1\right))\frac{n^2}{2}$. Then $G^{\prime }$ contains a $D{T}_2^t$ by Theorem 2 which contradicts the fact that $G^{\prime }$ contains no double edges.

Second we can assume that all but $o\left(n\right)$ vertices of G have weight at least $\frac{an}{2}(1+o\left(1\right))$. For otherwise let $v_1,\dots ,v_k,k=\lfloor ϵ·n\rfloor$ ($ϵ$ is a small positive number independent of n) be the vertices of G each of which has weight less than $\frac{an}{2}(1-c)$, where $0<c\left(ϵ\right)<c<1$. But then we have

$$\begin{array}{cc}\hfill e_a\left(G[v_{k+1},\dots ,v_n]\right)& \ge a(\frac{1}{2}+o\left(1\right))\frac{n^2}{2}-\frac{akn}{2}(1-c)\hfill \\ \hfill & =a(\frac{1}{4}(n^2-2kn+k^2)-\frac{k^2}{4}+\frac{ckn}{2}+\frac{n^2}{2}o\left(1\right))\hfill \\ \hfill & >\frac{a}{4}{(n-k)}^2(1+\delta (ϵ,c)),\hfill \end{array}$$

where $\delta (ϵ,c)>0$. By Theorem 2, we conclude that $G[v_{k+1},\dots ,v_n]$ contains a $T_3^t$ and so is G, which contradicts our assumption.

Now assume that $v_1,\dots ,v_p,p=(1+o\left(1\right))n$ be the vertices of G with weight not less than $\frac{an}{2}(1+o\left(1\right))$. Then the weight of each vertex of $G[v_1,\cdots ,v_p]$ in ($G[v_1,\cdots ,v_p]$) is at least $ap(\frac{1}{2}+o\left(1\right))=an(\frac{1}{2}+o\left(1\right))$. And $e_a\left(G[v_1,\cdots ,v_p]\right)=\frac{a{p}^2}{2}(\frac{1}{2}+o\left(1\right))=\frac{a{n}^2}{2}(\frac{1}{2}+o\left(1\right))$. Thus to prove our theorem it will suffice to show that $G[v_1,\cdots ,v_p]=D{T}_2\left(p\right)\pm o\left(p^2\right)$.

Thus it is clear that without loss of generality we can assume that the weight of every vertex in G is at least $an(\frac{1}{2}+o\left(1\right))$. Note that we now no longer have to use the assumption of $e_a\left(G\right)=\frac{a{n}^2}{2}(\frac{1}{2}+o\left(1\right))$. Since our assumption that $e_a\left(v_i\right)\ge an(\frac{1}{2}+o\left(1\right)),i=1,\dots ,n$ and G is $T_3^t-$free already implies that $e_a\left(G\right)=\frac{a{n}^2}{2}(\frac{1}{2}+o\left(1\right))$.

We shall show that if G is $T_3^t$-free digraph with $e_a\left(G\right)=\frac{a{n}^2}{2}(\frac{1}{2}+o\left(1\right))$ for some fixed t, then $G=D{T}_2\left(n\right)\pm o\left(n^2\right)$.

We call a double edge bad if it is contained in only $o\left(n\right)$ of $T_3$ in G. Assume first that G has at least $ϵn^2$ bad double edges. By Theorem 2, G contains $D{T}_2^t$ with two classes of vertices, say $u_1,\dots ,u_t$ and $v_1,\dots ,v_t$, so that all the double edges of $D{T}_2^t$ are bad. Now since the weight of each vertex in $\{u_1,\dots ,u_t,v_1,\dots ,v_t\}$ is at least $an(\frac{1}{2}+o\left(1\right))$ and each double edge $(u_i,v_j)(1\le i,j\le t)$ is contained in only $o\left(n\right)$ of $T_3$, a simple argument shows that the remaining $n-2t$ vertices of G can be divided into two classes (neglecting $o\left(n\right)$ vertices), say $z_1,\dots ,z_{u_1};w_1,\dots ,w_{u_2},u_1=(1+o\left(1\right))n/2,u_2=(1+o\left(1\right))n/2$, so that all the $u_i$ are connect to all the z’s with double edges and all the $y_j$ are connect to all the w’s.

If $e_a\left(G[z_1,\dots ,z_{u_1}]\right)=o\left(n^2\right)$ and $e_a\left(G[w_1,\dots ,w_{u_2}]\right)=o\left(n^2\right)$, then a simple computation shows that $D{T}_2(u_1,u_2)$ with the vertex set $\{z_1,\dots ,z_{u_1};w_1,\dots ,w_{u_2}\}$ differs from G by $o\left(n^2\right)$ edges, which prove our theorem (the remaining $n-u_1-u_2=o\left(n\right)$ vertices can be clearly ignored).

If, say $e_a\left(G[z_1,\dots ,z_{u_1}]\right)$ is not $o\left(n^2\right)$, then by Theorem 2 it contains a $D{T}_2^t$ with two classes of vertices, say $z_1,\dots ,z_t$ and $z_{t+1},\dots ,z_{2t}$. But then the digraph

$$\begin{array}{c}\hfill G[u_1,\dots ,u_t;z_1,\dots ,z_t;z_{t+1},\dots ,z_{2t}]\end{array}$$

clearly contains a $T_3^t$, which contradicts with our assumption.

Henceforth we can assume that there are $o\left(n^2\right)$ bad double edges. Bearing in mind that G contains $\mathsf{\Omega }\left(n^2\right)$ double edges. Let $e_1,\dots ,e_s,s>\alpha n^2$ be the double edges each of which is contained in at least $\beta n$ of $T_3$, where $\alpha ,\beta >0$. We now deduce from this assumption that G contains a $T_3^t$. Let $v_1^{\left(i\right)},\dots ,v_{r_i}^{\left(i\right)}$ be the vertices which form a $T_3$ with $e_i,r_i\ge \beta n,s\ge i\ge 1$. Since there are $2^r$ orientations of a star $S_{r+1}$ of $r+1$ vertices. Therefore there are at least ${\beta }^{\prime }n:=\beta n/2^r$ vertices of $\{v_j^{\left(i\right)},r_i\ge j\ge 1\}$ formed with $e_i$ with homogeneous $T_3$, say $\{v_j^{\left(i\right)},r_i^{\prime }\ge j\ge 1\},r_i^{\prime }\ge {\beta }^{\prime }n$ connect to both end vertices of $e_i$ in the same way. Similarly there are at least ${\alpha }^{\prime }{n}^2:=\alpha n^2/2^r$ double edges of $\{e_i,s\ge i\ge 1\}$ each formed with at least ${\beta }^{\prime }n$ vertices with homogeneous $T_3$. And all those $T_3$ formed with those at least ${\alpha }^{\prime }{n}^2$ double edges $e_i^{\prime }$ are homogeneous.

Form all possible t-tuple from those homogeneous vertices $v_{r_i}^{\left(i\right)}$. We get at least

$$\begin{array}{c}\hfill \sum _{i=1}^{{\alpha }^{\prime }{n}^2}\left(\genfrac{}{}{0pt}{}{r_i^{\prime }}{t}\right)\ge \sum _{i=1}^{{\alpha }^{\prime }{n}^2}\left(\genfrac{}{}{0pt}{}{{\beta }^{\prime }n}{t}\right)\ge {\alpha }^{\prime }{n}^2\frac{{\left({\beta }^{\prime }n\right)}^t}{3^{t}t!}>{\alpha }^{\prime }{n}^2{\left(\frac{{\beta }^{\prime }}{3}\right)}^t\left(\genfrac{}{}{0pt}{}{n}{t}\right)\end{array}$$

t-tuples. Since the total number of t-tuples formed from n elements is $\left(\genfrac{}{}{0pt}{}{n}{t}\right)$, there is a t-tuple say $z_1,\dots ,z_t$ which corresponds to at least ${\alpha }^{\prime }{n}^2{\left(\frac{{\beta }^{\prime }}{3}\right)}^t$ double edges $e_i$. By Theorem 2 these double edges determine a $D{T}_2^t$ with vertices $x_1,\dots ,x_t;y_1,\dots ,y_t$. Thus finally $G[x_1,\dots ,x_t$; $y_1,\dots ,y_t;z_1,\dots ,z_t]$ contains a $T_3^t$ as stated. But by assumption, G is $T_3^t$-free. This contradiction completes the proof. □

To keep all symbols consistent, we reshape Theorem 4 as follows:

Theorem of Stability.

Let $t\in \mathbb{N}$ and $a\in \mathbb{R}$ with $t\ge 1,a\in (\frac{3}{2},2]$. Then for any $\beta >0$ there exists $\gamma >0$ such that the following holds for all sufficiently large n. If a $T_3^t$-free digraph G on n vertices satisfies

$$\begin{array}{c}\hfill e_a\left(G\right)=a(\frac{1}{2}-\gamma )\frac{n^2}{2},\end{array}$$

then $G=D{T}_2\left(n\right)\pm \beta n^2$.

We also need the Digraph Removal Lemma of Alon and Shapira [18].

Lemma 6.

(Removal Lemma). For any fixed digraph H on h vertices, and any $\gamma >0$ there exists ${ϵ}^{\prime }>0$ such that the following holds for all sufficiently large n. If a digraph G on n vertices contains at most ${ϵ}^{\prime }{n}^h$ copies of H, then G can be made H-free by deleting at most $\gamma n^2$ edges.

Now we are ready to show that almost all $T_3^t$-free oriented graphs and almost all $T_3^t$-free digraphs are almost bipartite.

Proof of Theorem 3.

We only prove (i) here; the proof of (ii) is almost identical. Let $a:=log3$. Choose $n_0\in \mathbb{N}$ and $ϵ,\gamma ,\beta >0$ such that $1/n_0\ll ϵ\ll \gamma \ll \beta \ll \alpha ,1/r.$ Let ${ϵ}^{\prime }:=2ϵ$ and $n\ge n_0$. By Theorem 1 (with $T_3^t,n$ and $ϵ$ taking the roles of $H,N$ and $ϵ$ respectively) there is a collection $\mathcal{C}$ of digraphs on vertex set $\left[n\right]$ satisfying properties $\left(a\right)-\left(c\right)$. In particular, every $T_3^t$-free oriented graph on vertex set $\left[n\right]$ is contained in some digraph $G\in \mathcal{C}$. Let ${\mathcal{C}}_1$ be the family of all those $G\in \mathcal{C}$ for which $e_{log3}\left(G\right)\ge e{x}_{log3}(n,T_3^t)-{ϵ}^{\prime }{n}^2$. Then the number of $T_3^t$-free oriented graphs not contained in some $G\in {\mathcal{C}}_1$ is at most

$$\begin{array}{c}\hfill \left|\mathcal{C}\right|2^{e{x}_{log3}(n,T_3^t)-{ϵ}^{\prime }{n}^2}\le 2^{-ϵn^2}f(n,T_3^t),\end{array}$$

because $\left|\mathcal{C}\right|\le 2^{n^{2-{ϵ}^{\prime }}}$ and $f(n,T_3^t)\ge 2^{e{x}_{log3}(n,T_3^t)}$. Thus it suffices to show that every digraph $G\in {\mathcal{C}}_1$ satisfies $G=D{T}_2\left(n\right)\pm \alpha n^2$. By (b), each $G\in {\mathcal{C}}_1$ contains at most ${ϵ}^{\prime }{n}^{3t}$ copies of $T_3^t$. Thus by Lemma 6 we obtain a $T_3^t$-free digraph $G^{\prime }$ after deleting at most $\gamma n^2$ edges from G. Then $e_{log3}\left(G^{\prime }\right)\ge e{x}_{log3}(n,T_3^t)-({ϵ}^{\prime }+\gamma )n^2$. We next apply the Theorem of Stability to $G^{\prime }$ and derive that $G^{\prime }=D{T}_2\left(n\right)\pm \beta n^2$. As a result, the original digraph G satisfies $G=D{T}_2\left(n\right)\pm (\beta +\gamma )n^2$, hence $G=D{T}_2\left(n\right)\pm \alpha n^2$ as required. □

5. Concluding Remarks

In the work, we first give an analogue result of the Erdös-Stone theorem for weighted $T_{r+1}^t$-free digraphs. We then give a stability result of $T_3^t$-free oriented graphs and $T_3^t$-free digraphs. These results reconfirmed and strengthen Cherlin’s conjecture. However, we can not get the exact typical structures of $T_{r+1}^t$-free oriented graphs and digraphs. From our study experience and clues from other research, such as Kühn, Osthus, Townsend and Zhao [15], we believe that the exact structures are the same as those of $T_{r+1}$-free oriented graphs and digraphs. Therefore, we give the following conjecture at the end of this work:

Conjecture 2.

Let $r,t\in \mathbb{N}$ with $r\ge 2,t\ge 1$. Then the following hold.

(i) 

Almost all $T_{r+1}^t$-free oriented graph are r-partite.

(ii) 

Almost all $T_{r+1}^t$-free digraph are r-partite.