An Improved Error Term for Tur$\acute{\rm a}$n Number of Expanded Non-degenerate 2-graphs

TL;DR

This paper refines the upper bound for the Turán number of expanded non-degenerate 2-graphs, improving previous results and extending known cases where the extremal number equals a complete balanced partite r-graph.

Contribution

It provides a sharper error term for the Turán number of expanded 2-graphs, extending results to edge-critical graphs and improving bounds over Mubayi's 2016 work.

Findings

Improved error term for Turán number of expanded 2-graphs.

Shows equality with complete balanced partite r-graph for edge-critical F.

Extends previous bounds and results in extremal graph theory.

Abstract

For a 2-graph $F$, let $H_F^{(r)}$ be the $r$-graph obtained from $F$ by enlarging each edge with a new set of $r-2$ vertices. We show that if $\chi(F)=\ell>r \geq 2$, then $ {\rm ex}(n,H_F^{(r)})= t_r (n,\ell-1)+ \Theta( {\rm biex}(n,F)n^{r-2}),$ where $t_r (n,\ell-1)$ is the number of edges of an $n$-vertex complete balanced $\ell-1$ partite $r$-graph and ${\rm biex}(n,F)$ is the extremal number of the decomposition family of $F$. Since ${\rm biex}(n,F)=O(n^{2-\gamma})$ for some $\gamma>0$, this improves on the bound ${\rm ex}(n,H_F^{(r)})= t_r (n,\ell-1)+ o(n^r)$ by Mubayi (2016). Furthermore, our result implies that ${\rm ex}(n,H_F^{(r)})= t_r (n,\ell-1)$ when $F$ is edge-critical, which is an extension of the result of Pikhurko (2013).

Equations

$$\pi (H_{K_{\ell }}^{(r)})=\frac{r!}{(\ell -1{)}^r}\left(\genfrac{}{}{0ex}{}{\ell -1}{r}\right).$$

$$\pi (H_F^{(r)})=\frac{r!}{(\ell -1{)}^r}\left(\genfrac{}{}{0ex}{}{\ell -1}{r}\right).$$

$$\mathrm{biex}(n,F):=\mathrm{ex}(n,{\mathcal{F}}_F)=\mathrm{ex}(n,{\mathcal{F}}_F^{\star }).$$

$$\mathrm{ex}(n,H_F^{(r)})=t_r(n,\ell -1)+\Theta (\mathrm{biex}(n,F)n^{r-2}).$$

$$\mathrm{ex}(n,H_F^{(r)})=t_r(n,\ell -1).$$

$$p_G(x)=\sum _{\{i1,i2,\dots ,ir\}\in E}{x}_{i1}{x}_{i2}\cdots x_{ir},$$

$$\pi(F)=\sup\{r!\lambda(G):{\mbox{$G$ is $F$-hom-free}}\}.$$

$$\pi(F)=\sup_{\mbox{$G$ is $F$-hom-free}}r!\lambda(G)=r!\lambda(K_{\ell-1}^{r})=\frac{r!}{(\ell-1)^{r}}{(\ell-1)\choose r},$$

$$|G^{\prime }|\ge |G|-{\eta }_2\left(\genfrac{}{}{0ex}{}{n}{r}\right)\ge |T_r(n,\ell -1)|-\eta \left(\genfrac{}{}{0ex}{}{n}{r}\right)-{\eta }_2\left(\genfrac{}{}{0ex}{}{n}{r}\right)\ge |T_r(n,\ell -1)|-{\eta }_1\left(\genfrac{}{}{0ex}{}{n}{r}\right).$$

$$\alpha =\beta \pm \xi ,\text{mbox}if\beta -\xi \le \alpha \le \beta +\xi .$$

$$t_r(n,\ell -1)+c_1\mathrm{biex}(n,F)n^{r-2}\le \mathrm{ex}(n,H_F^{(r)})\le t_r(n,\ell -1)+c_2\mathrm{biex}(n,F)n^{r-2}.$$

$$G=T_r(n,\ell -1)\cup ({\cup }_{(u,v)\in H^{\prime }}E(u,v)),$$

$$|G|\ge t_r(n,\ell -1)+c\left(\genfrac{}{}{0ex}{}{\ell -2}{r-2}\right)\frac{1}{(\ell -1{)}^{r-2}}\mathrm{biex}(n,F)n^{r-2},$$

$$t_r(n,\ell -1)+c_1\mathrm{biex}(n,F)n^{r-2}\le \mathrm{ex}(n,H_F^{(r)}).$$

$$\kappa (n,F)=|V(H_F^{(r)})|\left(\genfrac{}{}{0ex}{}{n}{r-3}\right)$$

$$\delta (G^{\prime })\ge \delta (T_r(n,\ell -1)).$$

$$|G_{n-1}^{\prime }|$$

$$\displaystyle>|T_{r}(n,\ell-1)|-\delta(T_{r}(n,\ell-1))+\bigg{(}c_{2}{\rm biex}(n,F)n^{r-2}-{n\choose 2}\kappa(n,F)\bigg{)}$$

$$\ge |T_r(n-1,\ell -1)|+\frac{c_2}{2}\mathrm{biex}(n-1,F)(n-1{)}^{r-2}+1.$$

$$|G_{n-i}^{\prime }|\ge |T_r(n-i,\ell -1)|+\frac{c_2}{2^i}\mathrm{biex}(n-i,F)(n-i{)}^{r-2}+i.$$

$$E_{s,t}^{i,j}(v)=\{e\in C(v)\cup \bar{C}(v):|e\cap V_i|=s,|e\cap V_j|=t,|e\cap V_k|\le 1,k\ne i,j\}$$

$$|E_{2,0}^{i,j}(v)|\le |E_{1,1}^{i,j}(v)|j\ne i,j\in [\ell -1]$$

$$|V_i|=(1\pm \sqrt{\epsilon })\frac{n}{\ell -1},i\in [\ell -1]$$

$$\frac{1}{r}|X|{\epsilon }^{1/(4(r-1))}{n}^{r-1}\le 2\epsilon \left(\genfrac{}{}{0ex}{}{n}{r}\right),$$

$$d^c(v)$$

$$|N_{V_j}(v)\backslash X|$$

$$\ge (1-2(\ell -1{)}^{r-1}{\epsilon }^{\frac{1}{(4(r-1))}})\frac{n}{\ell -1}$$

$$\ge (1-{\epsilon }^{\frac{1}{(5(r-1))}})\frac{n}{\ell -1}$$

$$\frac{c_2}{2}\mathrm{biex}(n,F)n^{r-2}.$$

$$\sum _{i=1}^{\ell -1}|E(V_i^{\prime })|\ge \frac{c_2}{2}\mathrm{biex}(n,F)n^{r-2}-|X|n^{r-1}\ge \frac{c_2}{3}\mathrm{biex}(n,F)n^{r-2},$$

$$|E(V_{i^{\star }}^{\prime })|\ge \mathrm{biex}(n,F)n^{r-2}.$$

$$(1-\ell m{\epsilon }^{\frac{1}{(5(r-1))}})\frac{n}{\ell -1}-\ell m-|X|\ge (1-{\epsilon }^{\frac{1}{(6(r-1))}})\frac{n}{\ell -1}.$$

Full text

††journal: Journal of Graph Theory

1 Introduction

An $r$-graph (or $r$-uniform hypergraph) $G$ consists of a vertex set and an edge set with exactly $r$ vertices in each edge. We sometimes identify an $r$-graph $H$ with its edge set, and denote by $V(H)$ its vertex set. An $r$-clique of order $j$, denoted by $K_{j}^{(r)}$, is an $r$-graph on $j\geq r$ vertices consisting of all ${j\choose r}$ different $r$-tuples. An $r$-graph $G$ is said to be $k$-partite if its vertex set can be partitioned into $k$ classes $V_{1}\cup V_{2}\cup\cdots\cup V_{k}$ such that every edge of $G$ contains at most one vertex in $V_{i}$, $i=1,\ldots,k$. We say $G$ is a complete $k$-partite $r$-graph if $G$ consists of all $r$-tuples intersecting each vertex class in at most 1 vertex. Given two $r$-graphs $G$ and $F$, we say $G$ is $F$-free if $G$ does not have a (not necessarily induced) subgraph isomorphic to $F$. For a positive integer $\ell$, we denote by $[\ell]$ the set $\{1,\ldots,\ell\}$. For a set $V$ and an integer $r\geq 1$, let $[V]^{r}$ be the set of all $r$-element subsets of $V$. We write $[\ell]^{r}$ instead of $[[\ell]]^{r}$ for simple.

The Tur$\acute{\rm a}$n number ${\rm ex}(n,F)$ is the maximum number of edges in an $n$-vertex $F$-free $r$-graph. A simple and important averaging argument of Katona, Nemetz and Simonovits [3] shows that ${n\choose r}^{-1}{\rm ex}(n,F)$ form a decreasing sequence of real numbers in $[0,1]$. It follows that the sequence has a limit, called the Tur$\acute{\rm a}$n density and denoted by $\pi(F)$.

The Tur$\acute{\rm a}$n density is an asymptotic result ${\rm ex}(n,F)\sim\pi(F){n\choose r}$. An important fundamental theorem proved by Erd$\ddot{\rm o}$s and Stone [4] characterizes the Tur$\acute{\rm a}$n density of any 2-graph with its chromatic number.

Theorem 1.1** ([4]).**

Let $F$ be a 2-graph with $\chi(F)=\ell$, then $\pi(F)=\pi(K_{\ell})=\frac{\ell-2}{\ell-1}$.

However, when $F$ is an $r$-graph, $\pi(F)\neq 0$, and $r>2$, determining $\pi(F)$ is a hard problem, even for very simple $r$-graphs. In this paper, we focus on the non-degenerated expanded 2-graphs which are $r$-graphs defined as follows. Let $\ell>r\geq 2$ and let $F$ be a 2-graph, and $H_{F}^{(r)}$ be the $r$-graph obtained from $F$ by enlarging each edge with a new set of $r-2$ vertices. Mubayi [5] first determined the Tur$\acute{\rm a}$n density of expanded cliques and obtained a stability result.

Theorem 1.2** ([5]).**

Let $\ell>r\geq 2$. Then

[TABLE]

Later, using the stability method, Pikhurko [2] obtained the exact number of ${\rm ex}(n,H_{K_{\ell}}^{(r)})$.

It was mentioned in the survey of Mubayi [1] that Alon and Pikhurko observed that the approach applied to prove Theorem 2 in [2] can be extended to any edge-critical graph $F$ with $\chi(F)>r$. More generally, the following results can be easily achieved through a result of Erd$\ddot{\rm o}$s [6], the supersaturation technique (see Erd$\ddot{\rm o}$s-Simonovits [7]), and Theorem 1.2.

Theorem 1.3**.**

Let $\ell>r\geq 2$. Let $F$ be any 2-graph with $\chi(F)=\ell$, then

[TABLE]

This asymptotically gave the Tur$\acute{\rm a}$n number of all non-degenerated expanded 2-graphs.

For self completeness, we will give a short proof of Theorem 1.3 in the next section. Then we prove a stability result of $H_{F}^{(r)}$. Denote by $T_{r}(n,\ell)$ the complete $\ell$-partite $r$-graph on $n$ vertices, where the size of each vertex class differs at most 1, and set $t_{r}(n,\ell)=|T_{r}(n,\ell)|$. We say two $r$-graphs $G_{1}$ and $G_{2}$ of order $n$ are $\varepsilon$-close if we can add or remove at most $\varepsilon{n\choose r}$ edges from $G_{1}$ to make it isomorphic to $G_{2}$; in other words, for some bijection $\sigma:V(G_{1})\rightarrow V(G_{2})$ the symmetric difference between $\sigma(G_{1})=\{\sigma(D):D\in G_{1}\}$ and $G_{2}$ has at most $\varepsilon{n\choose r}$ edges.

Theorem 1.4** (Stability of $H_{F}^{(r)}$).**

Fix $\ell>r\geq 2$ and 2-graph $F$ with $\chi(F)=\ell$. For every $\varepsilon>0$, there exist $n_{0}=n_{0}(r,\ell,\varepsilon)>0$, $\eta=\eta(r,\ell,\varepsilon)>0$, such that if $n>n_{0}$ and $G$ is an $n$-vertex $H_{F}^{(r)}$-free $r$-graph with $|G|\geq|T_{r}(n,\ell-1)|-\eta{n\choose r}$, then $G$ is $\varepsilon$-close to $T_{r}(n,\ell-1)$.

Definition 1.1** (${\rm biex}(n,F)$).**

Given a 2-graph $F$ with $\chi(F)=\ell$, the decomposition family $\mathcal{F}_{F}$ of $F$ is the set of bipartite graphs which are obtained from $F$ by deleting $\ell-2$ colour classes in some $\ell$-colouring of $F$. Observe that $\mathcal{F}_{F}$ may contain graphs which are disconnected, or even have isolated vertices. Let $\mathcal{F}_{F}^{\star}$ be a minimal subfamily of $\mathcal{F}_{F}$ such that for any $H\in\mathcal{F}_{F}$, there exists $H^{\prime}\in\mathcal{F}_{F}^{\star}$ with $H^{\prime}\subset H$. We define

[TABLE]

Furthermore, we prove an improved bound for the Tur$\acute{\rm a}$n function ${\rm ex}(n,H_{F}^{(r)})$.

Theorem 1.5**.**

Given a $2$-graph $F$ with $\chi(F)=\ell>r$, then

[TABLE]

We say a 2-graph $F$ is edge-critical if there exists an edge $e\in E(F)$ such that $\chi(F)>\chi(F-e)$. The following theorem is a direct corollary of Theorem 1.5.

Theorem 1.6**.**

Given a $2$-graph $F$ with $\chi(F)=\ell>r$. If $F$ is edge-critical, then

[TABLE]

In the following section, we will a short proofs of Theorem 1.3 by using hypergraph Lagrange method. In section 3, we prove Theorem 1.4 based on the hypergraph removal lemma and a stability result of expanded cliques (see [2]). In the last section, we prove Theorem 1.5, the idea is first to identify a copy of $F$ in the ′2-shadow′ of an $r$-graph and then extend this copy to $H_{F}^{(r)}$.

2 Proof of Theorem 1.3

The hypergraph Lagrange method was developed independently by Sidorenko [8] and Frankl and F$\ddot{\rm u}$redi [9].

Let $G$ be an $r$-graph on $[n]=\{1,\ldots,n\}$ with edge set $E$, and $x=\{x_{1},\ldots,x_{n}\}$ with $x_{i}\geq 0$ for all $1\leq i\leq n$ and $\sum_{i=1}^{n}x_{i}=1$. Define

[TABLE]

The Lagrange of $G$ is defined as $\lambda(G)=\max_{x}p_{G}(x)$. $G$ is said to be dense if the inequality $\lambda(G^{\prime})<\lambda(G)$ holds for all its proper subgraphs $G^{\prime}\neq G$. We say that $G$ covers pairs if for every pair of vertices $i,j$ in $G$, there is an edge containing both $i$ and $j$. We need the following results.

Lemma 2.1** ([8],[9]).**

Every dense graph covers pairs.

Given $r$-graphs $F$ and $G$, we say $f:V(F)\rightarrow V(G)$ is a homomorphism if $f(e)\in E(G)$ for all $e\in E(F)$. And we call $G$ is $F$-hom-free if there is no homomorphism from $F$ to $G$. The following theorem shows how to compute the Tur$\acute{\rm a}$n density of any $r$-graph.

Lemma 2.2** ([8]).**

Let $F$ be an $r$-graph, then

[TABLE]

Proof 1  First, for the $(\ell-1)$-clique $K_{\ell-1}^{(r)}$, we have that $K_{\ell-1}^{(r)}$ is $H_{F}^{(r)}$-hom-free since $\chi(F)=\ell$. Otherwise, there is a homomorphism $f:F\rightarrow K_{\ell-1}^{(r)}$, then $f^{-1}(v_{i})$ form a vertex partition of $F$ and every edge of $H_{F}^{(r)}$ contains at most one vertex in $f^{-1}(v_{i})$. Thus $f^{-1}(v_{i})$ is an independent set in $F$, which is a contradiction.

On the other hand, for any dense graph $G$ on at least $\ell$ vertices, we can construct a homomorphism $g$ from $H_{F}^{(r)}$ to $G$. Since $\chi(F)=\ell$, so there is a partition of $V(F)$ into $\ell$ independent set. We map each independent set to $\ell$ distinct vertices of $G$. For the rest vertices in $H_{F}^{(r)}$, we denote the vertices in the edge containing $i,j$ by $v_{ij}^{k}$, $k=1,\ldots,r-2$. By Lemma 2.1, $G$ covers pairs. So there is an edge containing both $g(i)$ and $g(j)$. We map $v_{ij}^{k}$ to the rest $r-2$ distinct vertices in that edge. Thus $g$ is a homomorphism and by Lemma 2.2, we have

[TABLE]

which complete the proof. $\Box$

3 Stability of $H_{F}^{(r)}$

To proof Theorem 1.4, we need the following stability result of $H_{K_{\ell}}^{(r)}$ and the hypergraph removal lemma.

Lemma 3.1** ([2]).**

Fix $\ell>r\geq 2$. For every $\varepsilon_{1}>0$, there are $\eta_{1}=\eta_{1}(r,\ell,\varepsilon_{1})>0$ and $n_{0}=n_{0}(r,\ell,\varepsilon_{1})$ such that any $H_{K_{\ell}}^{(r)}$-free $r$-graph $G$ of order $n\geq n_{0}$ and size at least $|T_{r}(n,\ell-1)|-\eta_{1}n^{r}$ is $\varepsilon_{1}$-close to $T_{r}(n,\ell-1)$.

Hypergraph removal lemma is yield among a series extensions of the Szemer$\acute{\rm e}$di’s regularity lemma to $r$-graphs (see [10, 11, 12, 13]). Tao [14] also obtained such a generalization. In this paper, we will use two versions of the hypergraph removal lemma as follows.

Lemma 3.2** (Hypergraph Removal Lemma, [13]).**

Fix an $r$-graph $F$. For every $\varepsilon>0$, there exist $\eta>0$ and $n_{0}>0$ such that for every $n$-vertex $r$-graph $G$ with $n>n_{0}$, if $G$ contains at most $\eta n^{|V(F)|}$ copies of $F$, then one can delete at most $\varepsilon{n\choose r}$ edges to make it $F$-free.

The second version is as follows. The proof is also based on hypergraph regularity lemma and general dense counting lemma and similar to that of Lemma 3.2.

Lemma 3.3**.**

Fix an $r$-graph $F$. For every $\eta_{2}>0$, there exist $n_{0}>0$ such that for every $n$-vertex $r$-graph $G$ with $n>n_{0}$, if $G$ is $F$-free, then one can delete at most $\eta_{2}{n\choose r}$ edges to make it $F$-hom-free.

Proof of Theorem 1.4 We choose constant $\varepsilon_{1}+\eta_{2}<\varepsilon$ and $\eta+\eta_{2}<\eta_{1}$.

According to Lemma 3.3, we can delete at most $\eta_{2}{n\choose r}$ edges, denoting the remain $r$-graph by $G^{\prime}$, to make $G^{\prime}$ $H_{F}^{(r)}$-hom-free, which implies $G^{\prime}$ is $H_{K_{\ell}}^{(r)}$-free, and

[TABLE]

Apply Lemma 3.1 to $G^{\prime}$ for $\varepsilon_{1}$, we have $G^{\prime}$ is $\varepsilon_{1}$-close to $T_{r}(n,\ell-1)$. Thus $G$ is $(\varepsilon_{1}+\eta_{2})$-close to $T_{r}(n,\ell-1)$, which complete the proof. $\Box$

4 Proof of Theorem 1.5

For real constants $\alpha,\beta$, and a non-negative constant $\xi$, we write

[TABLE]

For $U\subseteq V(H^{(r)})$, we denote by $H^{(r)}[U]$ the sub-hypergraph of $H^{(r)}$ induced on $U$ (i.e. $H^{(r)}[U]=H^{(r)}\cap[U]^{r}$).

Given vertex sets $V_{1},\ldots,V_{\ell}$, let $K_{\ell}^{(j)}(V_{1},\ldots,V_{\ell})$ be the complete $\ell$-partite, $j$-graph. If $|V_{i}|=m$ for all $i\in[\ell]$, then an $(m,\ell,j)$-graph $H^{(j)}$ on $V_{1}\cup\cdots\cup V_{\ell}$ is any subset of $K_{\ell}^{(j)}(V_{1},\ldots,V_{\ell})$. Also, we regard the vertex partition $V_{1}\cup\cdots\cup V_{\ell}$ as an $(m,\ell,1)$-graph $H^{(1)}$. For $j\leq i\leq\ell$ and set $\Lambda_{i}\in[\ell]^{i}$, we denote the $\cup_{\lambda\in\Lambda_{i}}V_{\lambda}$ induced sub-hypergraph of the $(m,\ell,j)$-graph $H^{(j)}$ by $H^{(j)}[\Lambda_{i}]=H^{(j)}[\cup_{\lambda\in\Lambda_{i}}V_{\lambda}]$.

To prove Theorem 1.5, it is sufficient to prove the following theorem.

Theorem 4.1**.**

Given a $2$-graph $F$ with $\chi(F)=\ell>r\geq 2$, there exist $c_{1},c_{2},n_{0}>0$ such that if $n\geq n_{0}$, we have

[TABLE]

Proof of Theorem 4.1  Firstly, the left hand-side inequality is obtained as follows. Let $H$ be an $n$-vertex $\mathcal{F}_{F}$-free $2$-graph with ${\rm biex}(n,F)$ edges, and let $c=(\ell-1)^{-2}$, $c_{1}={{\ell-2}\choose{r-2}}\frac{c}{(\ell-1)^{r-2}}$. Obviously, there exists an $n/(\ell-1)$-vertex subgraph $H^{\prime}$ of $H$ with at least $c|H|$ edges.

Next, we construct $G$ from $T_{r}(n,\ell-1)$ as follows. Without loss of generality, let $V_{1}$ be the vertex class of $T_{r}(n,\ell-1)$ with largest size, we insert $H^{\prime}$ into $V_{1}$. Then for each edge $(u,v)$ in $H^{\prime}$, add all the $r$-tuples that contains $u,v$ and $(r-2)$ vertices chosen from different vertex classes except $V_{1}$ to $G$, i.e.,

[TABLE]

where $E(u,v)=\{\{u,v\}\cup f:|f|=r-2,|f\cap V_{i}|\leq 1,f\cap V_{1}=\emptyset,i=2\ldots,\ell-1\}$.

Clearly, we have

[TABLE]

and by definition of $\mathcal{F}_{F}$, the graph $G$ is $H_{F}^{(r)}$-free, and therefore

[TABLE]

Secondly, the main idea to prove the right hand-side inequality is to find a copy of $F$ in the 2-shadow of $G$, and then extend $F$ to $H_{F}^{(r)}$ in $G$. Here by saying 2-shadow of $G$, denoted by $\Delta_{2}(G)$, we mean the set of all 2-tuples $\{u,v\}\in[V(G)]^{2}$ that are contained in some edge of $G$.

Set $|V(F)|=m$ and choose $\varepsilon>0$ small enough. Suppose $G$ is an $n$-vertex $H_{F}^{(r)}$-free $r$-graph with $|G|>t_{r}(n,\ell-1)+c_{2}{\rm biex}(n,F)n^{r-2}$, $n\geq n_{0}$, then by Theorem 1.4 $G$ is $\varepsilon$-close to $T_{r}(n,\ell-1)$. Thus $V(G)$ can be partitioned into balanced $V_{1}\dot{\cup}\cdots\dot{\cup}V_{\ell-1}$ corresponding to $T_{r}(n,\ell-1)$.

Since $\frac{n}{2}\leq{\rm biex}(n,F)=o(n^{2})$ or ${\rm biex}(n,F)=0$, so we have

Fact 1. $\frac{1}{2}n^{r-1}<{\rm biex}(n,F)n^{r-2}=o(n^{r})$ or ${\rm biex}(n,F)n^{r-2}=0$.

We call a pair of vertices bad if it is covered by at most

[TABLE]

edges of $G$.

Let $G^{\prime}$ be obtained from $G$ by deleting all edges containing bad pairs, at most ${n\choose 2}\kappa(n,F)<\varepsilon{n\choose r}$. So $G^{\prime}$ is $2\varepsilon$-close to $T_{r}(n,\ell-1)$.

For any vertex $v$, we denote by $d(v)$ the vertex degree of $v$ in $G^{\prime}$, and denote by $N(v)$ the neighbours of $v$ in $G^{\prime}$, i.e., for each vertex $u$ in $N(v)$, there is an edge containing both $u$ and $v$. Let $N_{V_{i}}(v)=N(v)\cap V_{i}$ and $d_{i}(v)=|N_{V_{i}}(v)|$. An edge $e$ is crossing if $|e\cap V_{i}|\leq 1$ for $i\in[\ell-1]$. Let $C(v)$ be the set of crossing edges containing $v$ and $\bar{C}(v)$ be the set of non-crossing edges containing $v$, and we call $d^{c}(v)=|C(v)|$ the crossing degree of $v$ and $\bar{d}^{c}(v)=|\bar{C}(v)|$ the non-crossing degree.

Observe first that we may assume without loss of generality that

[TABLE]

where $\delta(G^{\prime})=\min\{d(v):v\in V(G^{\prime})\}$. Indeed, if this is not the case, we can repeatedly delete vertices of minimum degree of $G^{\prime}$ and delete all edges containing bad pairs until we arrive at a graph $G^{\prime}_{n^{\star}}$ on $n^{\star}$ vertices with $\delta(G_{n^{\star}})\geq\delta(T_{r}(n^{\star},\ell-1))$. Denote the sequence of graphs obtained in this way by $G^{\prime}_{n}:=G^{\prime},G^{\prime}_{n-1},\ldots,G^{\prime}_{n^{\star}}$. We need to verify that $n^{\star}\geq n_{0}$. Indeed, we have

[TABLE]

Similarly, we have

[TABLE]

Let $i^{\star}:=n-n^{\star}$. If $n^{\star}<n_{0}$, then $i^{\star}>n-n^{\frac{1}{2r}}$, which implies $i^{\star}\geq{{n-i^{\star}}\choose r}$, a contradiction. Hence we may assume (1).

Next we move the vertices to get a max $(\ell-1)$-cut of $G^{\prime}$, i.e., maximise the number of crossing edges. For $v\in V_{i}$ and $i\neq j\in[\ell-1]$, let

[TABLE]

Then, the max $(\ell-1)$-cut implies a vertex partition such that for each vertex $v\in V_{i}$, we have

[TABLE]

Since the number of crossing edges is at least $t_{r}(n,\ell-1)-2\varepsilon{n\choose r}$, so a simple computation would indicate that

[TABLE]

Note that $d^{c}(v)+\bar{d}^{c}(v)=d(v)$. Let $X=\{v\in V(G^{\prime}):\bar{d}^{c}(v)>\varepsilon^{1/(4(r-1))}n^{r-1}\}$ and $X_{i}=X\cap V_{i}$. Set $V^{\prime}_{i}=V_{i}\backslash X$. Since $G^{\prime}$ is $2\varepsilon$-close to $T_{r}(n,\ell-1)$, so

[TABLE]

which implies $|X|\leq\varepsilon^{2/3}n$.

Then for every $v\in V^{\prime}_{i}$, $i\in[\ell-1]$, we have

[TABLE]

and this implies that for every $j\neq i,j\in[\ell-1]$,

[TABLE]

Let $q$ be a positive constant depending only on $|V(F)|$ and $\varepsilon$. Its value will be given later.

Case 1. If $|X|<q(m-1)$ and ${\rm biex}(n,F)n^{r-2}>0$. Since $|G|>t_{r}(n,\ell-1)+c_{2}{\rm biex}(n,F)n^{r-2}$, so the number of non-crossing edges in $G^{\prime}$ is at least

[TABLE]

For every $i\in[\ell-1]$, we denote by $E(V^{\prime}_{i})$ the set of non-crossing edges in $G^{\prime}$ that contains at least 2 vertices in $V^{\prime}_{i}$. We have

[TABLE]

where the last inequality is due to Fact 1 and $n$ sufficiently large. Then, there exists some $i^{\star}$ such that

[TABLE]

Next we write $D=\{\{u,v\}\in[V_{i^{\star}}]^{2}:\mbox{there exists some }e\in E(V_{i^{\star}}),\ \mbox{such that }\{u,v\}\subseteq e\}$. Because each vertex pair in $D$ is contained in at most $n^{r-2}$ edges in $E(V_{i^{\star}})$, so, by (7), we have $|D|\geq{\rm biex}(n,F)$. That is, we can find some $H\in\mathcal{F}^{\star}_{F}$ in $\Delta_{2}(G^{\prime})[V^{\prime}_{i^{\star}}]$. Let such a copy of $H$ be fixed and assume without loss of generality that $V(H)\subseteq V_{\ell-1}$. Then we show that $H$ can be extended to a copy of $F$ in the 2-shadow of $G^{\prime}$ by finding a complete $(\ell-2)$-partite 2-graph in $\Delta_{2}(G^{\prime})$.

Note that by (5), we have for any vertex set $S\subseteq V(G)$ with $S\leq\ell m$ and every $i\in[\ell-1]$, the number of common neighbours in $V^{\prime}_{i}$ of every vertex in $S$ is at least

[TABLE]

The inequality is due to $\varepsilon$ is small enough and $n$ is sufficiently large.

We inductively find sets $S_{i}\subseteq V^{\prime}_{i}$ of size $m$ which form the parts of the complete $(\ell-2)$-partite 2-graph. For each $1\leq i\leq\ell-2$ in turn, we note that $|V(H)|+(i-1)m\leq\ell|V(H)|$, and therefore the set $V(H)\cup S_{1}\cup\cdots\cup S_{i-1}$ has at least $(1-\varepsilon^{\frac{1}{6(r-1)}})\frac{n}{\ell-1}\geq m$ common neighbours in $V^{\prime}_{i}$. We let $S_{i}$ be any set of size $m$ of these common neighbours. Hence we can extend $H$ to a copy of $F$ in $\Delta_{2}(G^{\prime})$.

Finally, recalling that we have deleted the edges that contains bad pairs, each vertex pair (or edge) in this copy of $F$ is contained in at least $(\ell+(r-2){\ell\choose 2}){n\choose{r-3}}$ edges of $G^{\prime}$. Thus we can choose, for each pair, one of these edges that is vertex-disjoint to the chosen ones to form a copy of $H_{F}^{(r)}$ in $G^{\prime}$.

Case 2. If $|X|\geq q(m-1)$ and ${\rm biex}(n,F)n^{r-2}>0$.

Let $\bar{C}_{i}(x)$ be the subset of $\bar{C}(x)$ that contains at 2 vertices in $V_{i}\backslash\{x\}$.

If there is $i,j\in[\ell-1]$ (Notice that $i=j$ is possible), and $x\in X_{i}$ such that $|\bar{C}_{j}(x)|\geq\sqrt{\varepsilon}n^{r-1}$.

We write $D=\{\{u,v\}\in[V_{j}]^{2}:\mbox{there exists some }e\in\bar{C}_{j}(x),\mbox{ s.t. }\{u,v\}\subseteq e\}$ and we claim $|D[V^{\prime}_{j}]|\geq{\rm biex}(n,F)$. Because the number of 2-tuples in $D$ that contains at least 1 vertex in $X_{i}$ is at most $\varepsilon^{\frac{2}{3}}n^{2}.$ Thus the number of edges in $\bar{C}_{j}(x)$ that contains at least 1 vertex in $X_{i}$ is at most $\varepsilon^{\frac{2}{3}}n^{2}n^{r-3}$. Note that each 2-tuple in $D[V^{\prime}_{j}]$ is contained in at most $n^{r-3}$ edges in $\bar{C}_{j}(x)$, thus

[TABLE]

This means that we can again find a copy of some $H\in\mathcal{F}_{F}$. And the extending from $H$ to $F$, and then to $H_{F}^{(r)}$ is the same as that in Case 1.

Otherwise for every $i,j\in[\ell-1]$, $x\in X_{i}$, we have $|\bar{C}_{j}(x)|\leq\sqrt{\varepsilon}n^{r-1}$. Denote by $\bar{C}_{i}^{1}(x)$ the subset of $\bar{C}(x)$ that contains exactly 1 vertex in $V_{i}\backslash x$. Then by $\sum_{j=1}^{\ell-1}|\bar{C}_{j}(x)|+|\bar{C}_{i}^{1}(x)|\geq\bar{d}^{c}(x)$, we know

[TABLE]

Claim 1. $|N(x)\cap V^{\prime}_{j}|\geq\varepsilon^{\frac{1}{2(r-1)}}n$ for every $j\in[\ell-1]$

Proof By (9), it is easy to know that

[TABLE]

Moveover, by (2), (9) and the assumption of $x$, we have for every $j\neq i$, $j\in[\ell-1]$

[TABLE]

So $|N(x)\cap V^{\prime}_{j}|\geq\varepsilon^{\frac{1}{2(r-1)}}n$ for $j\in[\ell-1]$. $\Box$

Now we start identifying a copy of $H_{F}^{(r)}$ in $G^{\prime}$ by 2 steps.

Set $X^{\prime}:=X$. The first step is to identify $m$ vertices in $X^{\prime}$ which are completely joined to an $(r-1)$-partite $r$-graph $H_{K_{\ell-1}(m)}^{(r)}$ in $V(G^{\prime})\backslash X$, with $m$ vertices, one vertex class of $K_{\ell-1}(m)$, in one vertex class of $V(G^{\prime})\backslash X$. The second step is to extent the structure identified in this way to a copy of $H_{F}^{(r)}$ in $G^{\prime}$, which is similar as that in Case 1.

By Claim 1, we can choose for each $i$ a set $S_{i}\in N(x)\cap V^{\prime}_{i}$ of size $\varepsilon^{\frac{1}{2(r-1)}}n$. Since $G^{\prime}$ is $2\varepsilon$-close to $T_{r}(n,\ell-1)$, so the graph $G^{\prime}[\dot{\cup}S_{i}]$ has density at least

[TABLE]

Then by Theorem 1.3, we can not remove any edge set of size $\varepsilon n^{r}$ to make $r$-graph $G^{\prime}[\dot{\cup}S_{i}]$ contain no $H_{K_{\ell-1}(m)}^{(r)}$. So by Lemma 3.2, there are at least $\eta n^{|V(H_{K_{\ell-1}(m)}^{(r)})|}$ copies of $H_{K_{\ell-1}(m)}^{(r)}$, where $\eta$ depends only on $m$ and $\varepsilon$. Choosing $q:=1/\eta$, we can then use the pigeonhole principle and the fact that $|X^{\prime}|>q(m-1)$ to infer that there are $m$ vertices in $|X^{\prime}|$ which are all adjacent to the vertices of one specific copy of $H_{K_{\ell-1}(m)}^{(r)}$ in $G^{\prime}[\dot{\cup}S_{i}]$ as desired.

Case 3. When ${\rm biex}(n,F)n^{r-2}=0$, i.e., the single edge graph $H\in\mathcal{F}_{F}$.

The only difference is that the condition (6) in Case 1 is no longer hold. We can change the proof slightly by using $G$ instead of $G^{\prime}$.

First, we change the assumption $|X|<q(m-1)$ of Case 1 to $X=\emptyset$. For $v\in V(G)$, We call the vertex $u$ a good neighbour of $v$ if $(u,v)$ are covered by at least $\kappa(n,F)$ edges of $G$. Note that, similar to (5), we still have the number of good neighbours in $V_{j}$ of $v\in V_{i}$ is at least $(1-\varepsilon^{\frac{1}{(5(r-1))}})\frac{n}{\ell-1}$. Except the only non-crossing edge we identify as a copy of $H$, the rest of proof in Case 1 is the same.

And in Case 2, only one vertex in $X$ is enough for us to find a copy of $H_{F}^{(r)}$ because there is a 1-vertex class in some coloring of $F$.

$\Box$